Elliptic curves over finite fields¶
AUTHORS:
William Stein (2005): Initial version
Robert Bradshaw et al….
John Cremona (2008-02): Point counting and group structure for non-prime fields, Frobenius endomorphism and order, elliptic logs
Mariah Lenox (2011-03): Added
set_ordermethodLorenz Panny, John Cremona (2023-02):
.twists()Lorenz Panny (2023):
special_supersingular_curve()Martin Grenouilloux, Gareth Ma (2024-09):
EllipticCurve_with_prime_order()
- class sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field(R, data, category=None)[source]¶
Bases:
EllipticCurve_fieldElliptic curve over a finite field.
EXAMPLES:
sage: EllipticCurve(GF(101),[2,3]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Finite Field of size 101 sage: # needs sage.rings.finite_rings sage: F = GF(101^2, 'a') sage: EllipticCurve([F(2),F(3)]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Finite Field in a of size 101^2
>>> from sage.all import * >>> EllipticCurve(GF(Integer(101)),[Integer(2),Integer(3)]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Finite Field of size 101 >>> # needs sage.rings.finite_rings >>> F = GF(Integer(101)**Integer(2), 'a') >>> EllipticCurve([F(Integer(2)),F(Integer(3))]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Finite Field in a of size 101^2
Elliptic curves over \(\ZZ/N\ZZ\) with \(N\) prime are of type “elliptic curve over a finite field”:
sage: F = Zmod(101) sage: EllipticCurve(F, [2, 3]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101 sage: E = EllipticCurve([F(2), F(3)]) sage: type(E) <class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'> sage: E.category() Category of abelian varieties over Ring of integers modulo 101
>>> from sage.all import * >>> F = Zmod(Integer(101)) >>> EllipticCurve(F, [Integer(2), Integer(3)]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101 >>> E = EllipticCurve([F(Integer(2)), F(Integer(3))]) >>> type(E) <class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'> >>> E.category() Category of abelian varieties over Ring of integers modulo 101
Elliptic curves over \(\ZZ/N\ZZ\) with \(N\) composite are of type “generic elliptic curve”:
sage: F = Zmod(95) sage: EllipticCurve(F, [2, 3]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95 sage: E = EllipticCurve([F(2), F(3)]) sage: type(E) <class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'> sage: E.category() Category of schemes over Ring of integers modulo 95 sage: TestSuite(E).run(skip=["_test_elements"])
>>> from sage.all import * >>> F = Zmod(Integer(95)) >>> EllipticCurve(F, [Integer(2), Integer(3)]) Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95 >>> E = EllipticCurve([F(Integer(2)), F(Integer(3))]) >>> type(E) <class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'> >>> E.category() Category of schemes over Ring of integers modulo 95 >>> TestSuite(E).run(skip=["_test_elements"])
- abelian_group()[source]¶
Return the abelian group structure of the group of points on this elliptic curve.
See also
If you do not need the complete abelian group structure but only generators of the group, use
gens()which can be much faster in some cases.This method relies on
gens(), which uses random points on the curve and hence the generators are likely to differ from one run to another. However, the group is cached, so the generators will not change in any one run of Sage.OUTPUT:
an
AdditiveAbelianGroupWrapperobject encapsulating the abelian group of rational points on this elliptic curve
ALGORITHM:
We first call
gens()to obtain a generating set \((P,Q)\). Letting \(P\) denote the point of larger order \(n_1\), we extend \(P\) to a basis \((P,Q')\) by computing a scalar \(x\) such that \(Q'=Q-[x]P\) has order \(n_2=\#E/n_1\). Finding \(x\) involves a (typically easy) discrete-logarithm computation.The complexity of the algorithm is the cost of factoring the group order, plus \(\Theta(\sqrt{\ell})\) for each prime \(\ell\) such that the rational \(\ell^\infty\)-torsion of
selfis isomorphic to \(\ZZ/\ell^r\times\ZZ/\ell^s\) with \(r>s>0\), times a polynomial in the logarithm of the base-field size.AUTHORS:
John Cremona: original implementation
Lorenz Panny (2021): current implementation
EXAMPLES:
sage: E = EllipticCurve(GF(11),[2,5]) sage: E.abelian_group() Additive abelian group isomorphic to Z/10 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x + 5 over Finite Field of size 11
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)),[Integer(2),Integer(5)]) >>> E.abelian_group() Additive abelian group isomorphic to Z/10 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x + 5 over Finite Field of size 11
sage: E = EllipticCurve(GF(41),[2,5]) sage: E.abelian_group() Additive abelian group isomorphic to Z/22 + Z/2 ...
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(41)),[Integer(2),Integer(5)]) >>> E.abelian_group() Additive abelian group isomorphic to Z/22 + Z/2 ...
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(3^6,'a') sage: E = EllipticCurve([a^4 + a^3 + 2*a^2 + 2*a, 2*a^5 + 2*a^3 + 2*a^2 + 1]) sage: E.abelian_group() Additive abelian group isomorphic to Z/26 + Z/26 ...
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(3)**Integer(6),'a', names=('a',)); (a,) = F._first_ngens(1) >>> E = EllipticCurve([a**Integer(4) + a**Integer(3) + Integer(2)*a**Integer(2) + Integer(2)*a, Integer(2)*a**Integer(5) + Integer(2)*a**Integer(3) + Integer(2)*a**Integer(2) + Integer(1)]) >>> E.abelian_group() Additive abelian group isomorphic to Z/26 + Z/26 ...
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(101^3,'a') sage: E = EllipticCurve([2*a^2 + 48*a + 27, 89*a^2 + 76*a + 24]) sage: E.abelian_group() Additive abelian group isomorphic to Z/1031352 ...
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(101)**Integer(3),'a', names=('a',)); (a,) = F._first_ngens(1) >>> E = EllipticCurve([Integer(2)*a**Integer(2) + Integer(48)*a + Integer(27), Integer(89)*a**Integer(2) + Integer(76)*a + Integer(24)]) >>> E.abelian_group() Additive abelian group isomorphic to Z/1031352 ...
The group can be trivial:
sage: E = EllipticCurve(GF(2), [0,0,1,1,1]) sage: E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + x + 1 over Finite Field of size 2
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(2)), [Integer(0),Integer(0),Integer(1),Integer(1),Integer(1)]) >>> E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + x + 1 over Finite Field of size 2
Of course, there are plenty of points if we extend the field:
sage: E.cardinality(extension_degree=100) 1267650600228231653296516890625
>>> from sage.all import * >>> E.cardinality(extension_degree=Integer(100)) 1267650600228231653296516890625
This tests the patch for Issue #3111, using 10 primes randomly selected:
sage: E = EllipticCurve('389a') sage: for p in [5927, 2297, 1571, 1709, 3851, 127, 3253, 5783, 3499, 4817]: ....: G = E.change_ring(GF(p)).abelian_group() sage: for p in prime_range(10000): # long time (19s on sage.math, 2011) ....: if p != 389: ....: G = E.change_ring(GF(p)).abelian_group()
>>> from sage.all import * >>> E = EllipticCurve('389a') >>> for p in [Integer(5927), Integer(2297), Integer(1571), Integer(1709), Integer(3851), Integer(127), Integer(3253), Integer(5783), Integer(3499), Integer(4817)]: ... G = E.change_ring(GF(p)).abelian_group() >>> for p in prime_range(Integer(10000)): # long time (19s on sage.math, 2011) ... if p != Integer(389): ... G = E.change_ring(GF(p)).abelian_group()
This tests that the bug reported in Issue #3926 has been fixed:
sage: # needs sage.rings.number_field sage: K.<i> = QuadraticField(-1) sage: OK = K.ring_of_integers() sage: P = K.factor(10007)[0][0] sage: OKmodP = OK.residue_field(P) sage: E = EllipticCurve([0, 0, 0, i, i + 3]) sage: Emod = E.change_ring(OKmodP); Emod Elliptic Curve defined by y^2 = x^3 + ibar*x + (ibar+3) over Residue field in ibar of Fractional ideal (10007) sage: Emod.abelian_group() #random generators (Multiplicative Abelian group isomorphic to C50067594 x C2, ((3152*ibar + 7679 : 7330*ibar + 7913 : 1), (8466*ibar + 1770 : 0 : 1)))
>>> from sage.all import * >>> # needs sage.rings.number_field >>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1) >>> OK = K.ring_of_integers() >>> P = K.factor(Integer(10007))[Integer(0)][Integer(0)] >>> OKmodP = OK.residue_field(P) >>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), i, i + Integer(3)]) >>> Emod = E.change_ring(OKmodP); Emod Elliptic Curve defined by y^2 = x^3 + ibar*x + (ibar+3) over Residue field in ibar of Fractional ideal (10007) >>> Emod.abelian_group() #random generators (Multiplicative Abelian group isomorphic to C50067594 x C2, ((3152*ibar + 7679 : 7330*ibar + 7913 : 1), (8466*ibar + 1770 : 0 : 1)))
- cardinality(algorithm=None, extension_degree=1)[source]¶
Return the number of points on this elliptic curve.
INPUT:
algorithm– (optional) string:'pari'– use the PARI C-library functionellcard'bsgs'– use the baby-step giant-step method asimplemented in Sage, with the Cremona-Sutherland version of Mestre’s trick
'exhaustive'– naive point counting'subfield'– reduce to a smaller field, provided that the j-invariant lies in a subfield'all'– compute cardinality with both'pari'and'bsgs'; return result if they agree or raise aAssertionErrorif they do not
extension_degree– integer \(d\) (default: 1); if the base field is \(\GF{q}\), return the cardinality ofselfover the extension \(\GF{q^d}\) of degree \(d\)
OUTPUT:
The order of the group of rational points of
selfover its base field, or over an extension field of degree \(d\) as above. The result is cached.EXAMPLES:
sage: # needs sage.rings.finite_rings sage: EllipticCurve(GF(4, 'a'), [1,2,3,4,5]).cardinality() 8 sage: k.<a> = GF(3^3) sage: l = [a^2 + 1, 2*a^2 + 2*a + 1, a^2 + a + 1, 2, 2*a] sage: EllipticCurve(k,l).cardinality() 29
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> EllipticCurve(GF(Integer(4), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 8 >>> k = GF(Integer(3)**Integer(3), names=('a',)); (a,) = k._first_ngens(1) >>> l = [a**Integer(2) + Integer(1), Integer(2)*a**Integer(2) + Integer(2)*a + Integer(1), a**Integer(2) + a + Integer(1), Integer(2), Integer(2)*a] >>> EllipticCurve(k,l).cardinality() 29
sage: # needs sage.rings.finite_rings sage: l = [1, 1, 0, 2, 0] sage: EllipticCurve(k, l).cardinality() 38
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> l = [Integer(1), Integer(1), Integer(0), Integer(2), Integer(0)] >>> EllipticCurve(k, l).cardinality() 38
An even bigger extension (which we check against Magma):
sage: # needs sage.rings.finite_rings sage: EllipticCurve(GF(3^100, 'a'), [1,2,3,4,5]).cardinality() 515377520732011331036459693969645888996929981504 sage: magma.eval("Order(EllipticCurve([GF(3^100)|1,2,3,4,5]))") # optional - magma '515377520732011331036459693969645888996929981504'
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> EllipticCurve(GF(Integer(3)**Integer(100), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 515377520732011331036459693969645888996929981504 >>> magma.eval("Order(EllipticCurve([GF(3^100)|1,2,3,4,5]))") # optional - magma '515377520732011331036459693969645888996929981504'
sage: EllipticCurve(GF(10007), [1,2,3,4,5]).cardinality() 10076 sage: EllipticCurve(GF(10007), [1,2,3,4,5]).cardinality(algorithm='pari') 10076 sage: EllipticCurve(GF(next_prime(10**20)), [1,2,3,4,5]).cardinality() 100000000011093199520
>>> from sage.all import * >>> EllipticCurve(GF(Integer(10007)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 10076 >>> EllipticCurve(GF(Integer(10007)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality(algorithm='pari') 10076 >>> EllipticCurve(GF(next_prime(Integer(10)**Integer(20))), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 100000000011093199520
The cardinality is cached:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(3^100, 'a'), [1,2,3,4,5]) sage: E.cardinality() is E.cardinality() True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(3)**Integer(100), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.cardinality() is E.cardinality() True
The following is very fast since the curve is actually defined over the prime field:
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(11^100) sage: E1 = EllipticCurve(k, [3,3]) sage: N1 = E1.cardinality(algorithm='subfield'); N1 137806123398222701841183371720896367762643312000384671846835266941791510341065565176497846502742959856128 sage: E1.cardinality_pari() == N1 True sage: E2 = E1.quadratic_twist() sage: N2 = E2.cardinality(algorithm='subfield'); N2 137806123398222701841183371720896367762643312000384656816094284101308193849980588362304472492174093035876 sage: E2.cardinality_pari() == N2 True sage: N1 + N2 == 2*(k.cardinality() + 1) True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(11)**Integer(100), names=('a',)); (a,) = k._first_ngens(1) >>> E1 = EllipticCurve(k, [Integer(3),Integer(3)]) >>> N1 = E1.cardinality(algorithm='subfield'); N1 137806123398222701841183371720896367762643312000384671846835266941791510341065565176497846502742959856128 >>> E1.cardinality_pari() == N1 True >>> E2 = E1.quadratic_twist() >>> N2 = E2.cardinality(algorithm='subfield'); N2 137806123398222701841183371720896367762643312000384656816094284101308193849980588362304472492174093035876 >>> E2.cardinality_pari() == N2 True >>> N1 + N2 == Integer(2)*(k.cardinality() + Integer(1)) True
We can count points over curves defined as a reduction:
sage: # needs sage.rings.number_field sage: x = polygen(QQ) sage: K.<w> = NumberField(x^2 + x + 1) sage: EK = EllipticCurve(K, [0, 0, w, 2, 1]) sage: E = EK.base_extend(K.residue_field(2)) sage: E Elliptic Curve defined by y^2 + wbar*y = x^3 + 1 over Residue field in wbar of Fractional ideal (2) sage: E.cardinality() 7 sage: E = EK.base_extend(K.residue_field(w - 1)) sage: E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 2*x + 1 over Residue field of Fractional ideal (w - 1)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> x = polygen(QQ) >>> K = NumberField(x**Integer(2) + x + Integer(1), names=('w',)); (w,) = K._first_ngens(1) >>> EK = EllipticCurve(K, [Integer(0), Integer(0), w, Integer(2), Integer(1)]) >>> E = EK.base_extend(K.residue_field(Integer(2))) >>> E Elliptic Curve defined by y^2 + wbar*y = x^3 + 1 over Residue field in wbar of Fractional ideal (2) >>> E.cardinality() 7 >>> E = EK.base_extend(K.residue_field(w - Integer(1))) >>> E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 2*x + 1 over Residue field of Fractional ideal (w - 1)
sage: R.<x> = GF(17)[] sage: pol = R.irreducible_element(5) sage: k.<a> = R.residue_field(pol) sage: E = EllipticCurve(R, [1, x]).base_extend(k) sage: E Elliptic Curve defined by y^2 = x^3 + x + a over Residue field in a of Principal ideal (x^5 + x + 14) of Univariate Polynomial Ring in x over Finite Field of size 17 sage: E.cardinality() 1421004
>>> from sage.all import * >>> R = GF(Integer(17))['x']; (x,) = R._first_ngens(1) >>> pol = R.irreducible_element(Integer(5)) >>> k = R.residue_field(pol, names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(R, [Integer(1), x]).base_extend(k) >>> E Elliptic Curve defined by y^2 = x^3 + x + a over Residue field in a of Principal ideal (x^5 + x + 14) of Univariate Polynomial Ring in x over Finite Field of size 17 >>> E.cardinality() 1421004
- cardinality_bsgs(verbose=False)[source]¶
Return the cardinality of
selfover the base field.ALGORITHM: A variant of “Mestre’s trick” extended to all finite fields by Cremona and Sutherland, 2008.
Note
The Mestre-Schoof-Cremona-Sutherland algorithm may fail for a small finite number of curves over \(F_q\) for \(q\) at most 49, so for \(q<50\) we use an exhaustive count.
Quadratic twists are not implemented in characteristic 2 when \(j=0 (=1728)\); but this case is treated separately.
EXAMPLES:
sage: p = next_prime(10^3) sage: E = EllipticCurve(GF(p), [3,4]) sage: E.cardinality_bsgs() 1020 sage: E = EllipticCurve(GF(3^4,'a'), [1,1]) sage: E.cardinality_bsgs() 64 sage: F.<a> = GF(101^3,'a') sage: E = EllipticCurve([2*a^2 + 48*a + 27, 89*a^2 + 76*a + 24]) sage: E.cardinality_bsgs() 1031352
>>> from sage.all import * >>> p = next_prime(Integer(10)**Integer(3)) >>> E = EllipticCurve(GF(p), [Integer(3),Integer(4)]) >>> E.cardinality_bsgs() 1020 >>> E = EllipticCurve(GF(Integer(3)**Integer(4),'a'), [Integer(1),Integer(1)]) >>> E.cardinality_bsgs() 64 >>> F = GF(Integer(101)**Integer(3),'a', names=('a',)); (a,) = F._first_ngens(1) >>> E = EllipticCurve([Integer(2)*a**Integer(2) + Integer(48)*a + Integer(27), Integer(89)*a**Integer(2) + Integer(76)*a + Integer(24)]) >>> E.cardinality_bsgs() 1031352
- cardinality_exhaustive()[source]¶
Return the cardinality of
selfover the base field.This simply adds up the number of points with each x-coordinate: only used for small field sizes!
EXAMPLES:
sage: p = next_prime(10^3) sage: E = EllipticCurve(GF(p), [3,4]) sage: E.cardinality_exhaustive() 1020 sage: E = EllipticCurve(GF(3^4,'a'), [1,1]) sage: E.cardinality_exhaustive() 64
>>> from sage.all import * >>> p = next_prime(Integer(10)**Integer(3)) >>> E = EllipticCurve(GF(p), [Integer(3),Integer(4)]) >>> E.cardinality_exhaustive() 1020 >>> E = EllipticCurve(GF(Integer(3)**Integer(4),'a'), [Integer(1),Integer(1)]) >>> E.cardinality_exhaustive() 64
- cardinality_pari()[source]¶
Return the cardinality of
selfusing PARI.This uses pari:ellcard.
EXAMPLES:
sage: p = next_prime(10^3) sage: E = EllipticCurve(GF(p),[3,4]) sage: E.cardinality_pari() 1020 sage: K = GF(next_prime(10^6)) sage: E = EllipticCurve(K,[1,0,0,1,1]) sage: E.cardinality_pari() 999945
>>> from sage.all import * >>> p = next_prime(Integer(10)**Integer(3)) >>> E = EllipticCurve(GF(p),[Integer(3),Integer(4)]) >>> E.cardinality_pari() 1020 >>> K = GF(next_prime(Integer(10)**Integer(6))) >>> E = EllipticCurve(K,[Integer(1),Integer(0),Integer(0),Integer(1),Integer(1)]) >>> E.cardinality_pari() 999945
Since Issue #16931, this now works over finite fields which are not prime fields:
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(7^3) sage: E = EllipticCurve_from_j(a) sage: E.cardinality_pari() 318 sage: K.<a> = GF(3^20) sage: E = EllipticCurve(K,[1,0,0,1,a]) sage: E.cardinality_pari() 3486794310
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(7)**Integer(3), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve_from_j(a) >>> E.cardinality_pari() 318 >>> K = GF(Integer(3)**Integer(20), names=('a',)); (a,) = K._first_ngens(1) >>> E = EllipticCurve(K,[Integer(1),Integer(0),Integer(0),Integer(1),a]) >>> E.cardinality_pari() 3486794310
- count_points(n=1)[source]¶
Return the cardinality of this elliptic curve over the base field or extensions.
INPUT:
n– positive integer
OUTPUT: if \(n=1\), returns the cardinality of the curve over its base field
If \(n>1\), returns a list \([c_1, c_2, ..., c_n]\) where \(c_d\) is the cardinality of the curve over the extension of degree \(d\) of its base field.
EXAMPLES:
sage: p = 101 sage: F = GF(p) sage: E = EllipticCurve(F, [2,3]) sage: E.count_points(1) 96 sage: E.count_points(5) [96, 10368, 1031904, 104053248, 10509895776]
>>> from sage.all import * >>> p = Integer(101) >>> F = GF(p) >>> E = EllipticCurve(F, [Integer(2),Integer(3)]) >>> E.count_points(Integer(1)) 96 >>> E.count_points(Integer(5)) [96, 10368, 1031904, 104053248, 10509895776]
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(p^2) sage: E = EllipticCurve(F, [a,a]) sage: E.cardinality() 10295 sage: E.count_points() 10295 sage: E.count_points(1) 10295 sage: E.count_points(5) [10295, 104072155, 1061518108880, 10828567126268595, 110462212555439192375]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(p**Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> E = EllipticCurve(F, [a,a]) >>> E.cardinality() 10295 >>> E.count_points() 10295 >>> E.count_points(Integer(1)) 10295 >>> E.count_points(Integer(5)) [10295, 104072155, 1061518108880, 10828567126268595, 110462212555439192375]
- endomorphism_discriminant_from_class_number(h)[source]¶
Return the endomorphism order discriminant of this ordinary elliptic curve, given its class number
h.INPUT:
h– positive integer
OUTPUT:
integer; the discriminant of the endomorphism ring \(\text{End}(E)\), if this has class number
h. If \(\text{End}(E)\) does not have class numberh, aValueErroris raised.ALGORITHM:
Compute the trace of Frobenius and hence the discriminant \(D_0\) and class number \(h_0\) of the maximal order containing the endomorphism order. From the given value of \(h\), which must be a multiple of \(h_0\), compute the possible conductors, using
height_above_floor()for each prime \(\ell\) dividing the quotient \(h/h_0\). If exactly one conductor \(f\) remains, return \(f^2D_0\), otherwise raise aValueError; this can onlyhappen when the input value of \(h\) was incorrect.Note
Adapted from [RouSuthZur2022]. The application for which one knows the class number in advance is in the recognition of Hilbert Class Polynomials: see
sage.schemes.elliptic_curves.cm.is_HCP().EXAMPLES:
sage: F = GF(312401) sage: E = EllipticCurve(F,(0, 0, 0, 309381, 93465)) sage: E.endomorphism_discriminant_from_class_number(30) -671
>>> from sage.all import * >>> F = GF(Integer(312401)) >>> E = EllipticCurve(F,(Integer(0), Integer(0), Integer(0), Integer(309381), Integer(93465))) >>> E.endomorphism_discriminant_from_class_number(Integer(30)) -671
We check that this is the correct discriminant, and the input value of \(h\) was correct:
sage: H = hilbert_class_polynomial(-671) sage: H(E.j_invariant()) == 0 and H.degree()==30 True
>>> from sage.all import * >>> H = hilbert_class_polynomial(-Integer(671)) >>> H(E.j_invariant()) == Integer(0) and H.degree()==Integer(30) True
- frobenius()[source]¶
Return the frobenius of
selfas an element of a quadratic order.Note
This computes the curve cardinality, which may be time-consuming.
Frobenius is only determined up to conjugacy.
EXAMPLES:
sage: E = EllipticCurve(GF(11),[3,3]) sage: E.frobenius() phi sage: E.frobenius().minpoly() x^2 - 4*x + 11
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)),[Integer(3),Integer(3)]) >>> E.frobenius() phi >>> E.frobenius().minpoly() x^2 - 4*x + 11
For some supersingular curves, Frobenius is in Z:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(25,'a'),[0,0,0,0,1]) sage: E.frobenius() -5
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(25),'a'),[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)]) >>> E.frobenius() -5
- frobenius_discriminant()[source]¶
Return the discriminant of the ring \(\ZZ[\pi_E]\) where \(\pi_E\) is the Frobenius endomorphism.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<t> = GF(11^4) sage: E = EllipticCurve([t,t]) sage: E.frobenius_discriminant() -57339
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(11)**Integer(4), names=('t',)); (t,) = F._first_ngens(1) >>> E = EllipticCurve([t,t]) >>> E.frobenius_discriminant() -57339
- frobenius_endomorphism()[source]¶
Return the \(q\)-power Frobenius endomorphism of this elliptic curve, where \(q\) is the cardinality of the (finite) base field.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<t> = GF(11^4) sage: E = EllipticCurve([t,t]) sage: E.frobenius_endomorphism() Frobenius endomorphism of degree 14641 = 11^4: From: Elliptic Curve defined by y^2 = x^3 + t*x + t over Finite Field in t of size 11^4 To: Elliptic Curve defined by y^2 = x^3 + t*x + t over Finite Field in t of size 11^4 sage: E.frobenius_endomorphism() == E.frobenius_isogeny(4) True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(11)**Integer(4), names=('t',)); (t,) = F._first_ngens(1) >>> E = EllipticCurve([t,t]) >>> E.frobenius_endomorphism() Frobenius endomorphism of degree 14641 = 11^4: From: Elliptic Curve defined by y^2 = x^3 + t*x + t over Finite Field in t of size 11^4 To: Elliptic Curve defined by y^2 = x^3 + t*x + t over Finite Field in t of size 11^4 >>> E.frobenius_endomorphism() == E.frobenius_isogeny(Integer(4)) True
See also
- frobenius_order()[source]¶
Return the quadratic order Z[phi] where phi is the Frobenius endomorphism of the elliptic curve.
Note
This computes the curve cardinality, which may be time-consuming.
EXAMPLES:
sage: E = EllipticCurve(GF(11),[3,3]) sage: E.frobenius_order() Order of conductor 2 generated by phi in Number Field in phi with defining polynomial x^2 - 4*x + 11
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)),[Integer(3),Integer(3)]) >>> E.frobenius_order() Order of conductor 2 generated by phi in Number Field in phi with defining polynomial x^2 - 4*x + 11
For some supersingular curves, Frobenius is in Z and the Frobenius order is Z:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(25,'a'),[0,0,0,0,1]) sage: R = E.frobenius_order() sage: R Order generated by [] in Number Field in phi with defining polynomial x + 5 sage: R.degree() 1
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(25),'a'),[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)]) >>> R = E.frobenius_order() >>> R Order generated by [] in Number Field in phi with defining polynomial x + 5 >>> R.degree() 1
- frobenius_polynomial()[source]¶
Return the characteristic polynomial of Frobenius.
The Frobenius endomorphism of the elliptic curve has quadratic characteristic polynomial. In most cases this is irreducible and defines an imaginary quadratic order; for some supersingular curves, Frobenius is an integer a and the polynomial is \((x-a)^2\).
Note
This computes the curve cardinality, which may be time-consuming.
EXAMPLES:
sage: E = EllipticCurve(GF(11),[3,3]) sage: E.frobenius_polynomial() x^2 - 4*x + 11
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)),[Integer(3),Integer(3)]) >>> E.frobenius_polynomial() x^2 - 4*x + 11
For some supersingular curves, Frobenius is in Z and the polynomial is a square:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(25,'a'),[0,0,0,0,1]) sage: E.frobenius_polynomial().factor() (x + 5)^2
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(25),'a'),[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)]) >>> E.frobenius_polynomial().factor() (x + 5)^2
- gens()[source]¶
Return points which generate the abelian group of points on this elliptic curve.
The algorithm involves factoring the group order of
self, but is otherwise (randomized) polynomial-time.(The points returned by this function are not guaranteed to be the same each time, although they should remain fixed within a single run of Sage unless
abelian_group()is called.)OUTPUT: a tuple of points on the curve
if the group is trivial: an empty tuple.
if the group is cyclic: a tuple with 1 point, a generator.
if the group is not cyclic: a tuple with 2 points, where the order of the first point equals the exponent of the group.
Warning
In the case of 2 generators \(P\) and \(Q\), it is not guaranteed that the group is the cartesian product of the 2 cyclic groups \(\langle P \rangle\) and \(\langle Q \rangle\). In other words, the order of \(Q\) is not as small as possible. If you really need a basis (rather than just a generating set) of the group, use
abelian_group().EXAMPLES:
sage: E = EllipticCurve(GF(11),[2,5]) sage: P = E.gens()[0]; P # random (0 : 7 : 1) sage: E.cardinality(), P.order() (10, 10) sage: E = EllipticCurve(GF(41),[2,5]) sage: E.gens() # random ((20 : 38 : 1), (25 : 31 : 1)) sage: E.cardinality() 44
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)),[Integer(2),Integer(5)]) >>> P = E.gens()[Integer(0)]; P # random (0 : 7 : 1) >>> E.cardinality(), P.order() (10, 10) >>> E = EllipticCurve(GF(Integer(41)),[Integer(2),Integer(5)]) >>> E.gens() # random ((20 : 38 : 1), (25 : 31 : 1)) >>> E.cardinality() 44
If the abelian group has been computed, return those generators instead:
sage: E.abelian_group() Additive abelian group isomorphic to Z/22 + Z/2 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x + 5 over Finite Field of size 41 sage: ab_gens = E.abelian_group().gens() sage: ab_gens == E.gens() True sage: E.gens()[0].order() 22 sage: E.gens()[1].order() 2
>>> from sage.all import * >>> E.abelian_group() Additive abelian group isomorphic to Z/22 + Z/2 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x + 5 over Finite Field of size 41 >>> ab_gens = E.abelian_group().gens() >>> ab_gens == E.gens() True >>> E.gens()[Integer(0)].order() 22 >>> E.gens()[Integer(1)].order() 2
Examples with 1 and 0 generators:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(3^6) sage: E = EllipticCurve([a, a+1]) sage: pts = E.gens() sage: len(pts) 1 sage: pts[0].order() == E.cardinality() True sage: E = EllipticCurve(GF(2), [0,0,1,1,1]) sage: E.gens() ()
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(3)**Integer(6), names=('a',)); (a,) = F._first_ngens(1) >>> E = EllipticCurve([a, a+Integer(1)]) >>> pts = E.gens() >>> len(pts) 1 >>> pts[Integer(0)].order() == E.cardinality() True >>> E = EllipticCurve(GF(Integer(2)), [Integer(0),Integer(0),Integer(1),Integer(1),Integer(1)]) >>> E.gens() ()
This works over larger finite fields where
abelian_group()may be too expensive:sage: # needs sage.rings.finite_rings sage: k.<a> = GF(5^60) sage: E = EllipticCurve([a, a]) sage: len(E.gens()) 2 sage: E.cardinality() 867361737988403547206134229616487867594472 sage: a = E.gens()[0].order(); a # random 433680868994201773603067114808243933797236 sage: b = E.gens()[1].order(); b # random 30977204928157269543076222486303138128374 sage: lcm(a,b) 433680868994201773603067114808243933797236
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(5)**Integer(60), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve([a, a]) >>> len(E.gens()) 2 >>> E.cardinality() 867361737988403547206134229616487867594472 >>> a = E.gens()[Integer(0)].order(); a # random 433680868994201773603067114808243933797236 >>> b = E.gens()[Integer(1)].order(); b # random 30977204928157269543076222486303138128374 >>> lcm(a,b) 433680868994201773603067114808243933797236
- has_order(value, num_checks=8)[source]¶
Return
Trueif the curve has ordervalue.INPUT:
value– integer in the Hasse-Weil range for this curvenum_checks– integer (default: \(8\)); the number of times to check whethervaluetimes a random point on this curve equals the identity. Ifvalueis a prime and the curve is over a field of order at least \(5\), it is sufficient to pass innum_checks=1- see the examples below.
Note
Since the method is probabilistic, there is a possibility for the method to yield false positives (i.e. returning
Trueeven when the result isFalse). Even worse, it is possible for this to happen even whennum_checksis increased arbitrarily. See below for an example and Issue #38617 for an open discussion.EXAMPLES:
For curves over small finite fields, the order is computed and compared directly:
sage: E = EllipticCurve(GF(7), [0, 1]) sage: E.order() 12 sage: E.has_order(12, num_checks=0) True sage: E.has_order(11, num_checks=0) False sage: E.has_order(13, num_checks=0) False
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(7)), [Integer(0), Integer(1)]) >>> E.order() 12 >>> E.has_order(Integer(12), num_checks=Integer(0)) True >>> E.has_order(Integer(11), num_checks=Integer(0)) False >>> E.has_order(Integer(13), num_checks=Integer(0)) False
This tests the method on a random curve:
sage: # long time (10s) sage: p = random_prime(2**128, lbound=2**127) sage: K = GF((p, 2), name="a") sage: E = EllipticCurve(K, [K.random_element() for _ in range(2)]) sage: N = E.order() sage: E.has_order(N, num_checks=20) True sage: E.has_order(N + 1) False
>>> from sage.all import * >>> # long time (10s) >>> p = random_prime(Integer(2)**Integer(128), lbound=Integer(2)**Integer(127)) >>> K = GF((p, Integer(2)), name="a") >>> E = EllipticCurve(K, [K.random_element() for _ in range(Integer(2))]) >>> N = E.order() >>> E.has_order(N, num_checks=Integer(20)) True >>> E.has_order(N + Integer(1)) False
This demonstrates the bug mentioned in the NOTE above. The last return value should be
Falseafter Issue #38617 is fixed:sage: E = EllipticCurve(GF(5443568676570036275321323), [0, 13]) sage: N = 2333145661241 sage: E.order() == N^2 True sage: E.has_order(N^2) True sage: del E._order sage: E.has_order(N^2 + N) True
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(5443568676570036275321323)), [Integer(0), Integer(13)]) >>> N = Integer(2333145661241) >>> E.order() == N**Integer(2) True >>> E.has_order(N**Integer(2)) True >>> del E._order >>> E.has_order(N**Integer(2) + N) True
Due to the nature of the algorithm (testing multiple of points) and the Hasse-Weil bound, we see that for testing prime orders,
num_checks=1is sufficient:sage: p = random_prime(1000) sage: E = EllipticCurve(GF(p), j=randrange(p)) sage: q = random_prime(p + 20, lbound=p - 20) sage: E.has_order(q, num_checks=20) == E.has_order(q, num_checks=1) True
>>> from sage.all import * >>> p = random_prime(Integer(1000)) >>> E = EllipticCurve(GF(p), j=randrange(p)) >>> q = random_prime(p + Integer(20), lbound=p - Integer(20)) >>> E.has_order(q, num_checks=Integer(20)) == E.has_order(q, num_checks=Integer(1)) True
AUTHORS:
Mariah Lenox (2011-02-16): Initial implementation
Gareth Ma (2024-01-21): Fix bug for small curves
- height_above_floor(ell, e)[source]¶
Return the height of the \(j\)-invariant of this ordinary elliptic curve on its \(\ell\)-volcano.
INPUT:
ell– a prime numbere– nonnegative integer, the \(\ell\)-adic valuation of the conductor the Frobenius order
Note
For an ordinary \(E/\GF{q}\), and a prime \(\ell\), the height \(e\) of the \(\ell\)-volcano containing \(j(E)\) is the \(\ell\)-adic valuation of the conductor of the order generated by the Frobenius \(\pi_E\); the height of \(j(E)\) on its ell-volcano is the \(\ell\)-adic valuation of the conductor of the order \(\text{End}(E)\).
ALGORITHM:
See [RouSuthZur2022].
EXAMPLES:
sage: F = GF(312401) sage: E = EllipticCurve(F,(0, 0, 0, 309381, 93465)) sage: D = E.frobenius_discriminant(); D -687104 sage: D.factor() -1 * 2^10 * 11 * 61 sage: E.height_above_floor(2,8) 5
>>> from sage.all import * >>> F = GF(Integer(312401)) >>> E = EllipticCurve(F,(Integer(0), Integer(0), Integer(0), Integer(309381), Integer(93465))) >>> D = E.frobenius_discriminant(); D -687104 >>> D.factor() -1 * 2^10 * 11 * 61 >>> E.height_above_floor(Integer(2),Integer(8)) 5
- is_isogenous(other, field=None, proof=True)[source]¶
Return whether or not
selfis isogenous toother.INPUT:
other– another elliptic curvefield– (default:None) a field containing the base fields of the two elliptic curves into which the two curves may be extended to test if they are isogenous over this field. By default is_isogenous will not try to find this field unless one of the curves can be extended into the base field of theother, in which case it will test over the larger base field.proof– boolean (default:True); this parameter is here only to be consistent with versions for other types of elliptic curves
OUTPUT:
boolean;
Trueif there is an isogeny from curveselfto curveotherdefined overfieldEXAMPLES:
sage: # needs sage.rings.finite_rings sage: E1 = EllipticCurve(GF(11^2,'a'),[2,7]); E1 Elliptic Curve defined by y^2 = x^3 + 2*x + 7 over Finite Field in a of size 11^2 sage: E1.is_isogenous(5) Traceback (most recent call last): ... ValueError: Second argument is not an Elliptic Curve. sage: E1.is_isogenous(E1) True sage: # needs sage.rings.finite_rings sage: E2 = EllipticCurve(GF(7^3,'b'),[3,1]); E2 Elliptic Curve defined by y^2 = x^3 + 3*x + 1 over Finite Field in b of size 7^3 sage: E1.is_isogenous(E2) Traceback (most recent call last): ... ValueError: The base fields must have the same characteristic. sage: # needs sage.rings.finite_rings sage: E3 = EllipticCurve(GF(11^2,'c'),[4,3]); E3 Elliptic Curve defined by y^2 = x^3 + 4*x + 3 over Finite Field in c of size 11^2 sage: E1.is_isogenous(E3) False sage: # needs sage.rings.finite_rings sage: E4 = EllipticCurve(GF(11^6,'d'),[6,5]); E4 Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in d of size 11^6 sage: E1.is_isogenous(E4) True sage: # needs sage.rings.finite_rings sage: E5 = EllipticCurve(GF(11^7,'e'),[4,2]); E5 Elliptic Curve defined by y^2 = x^3 + 4*x + 2 over Finite Field in e of size 11^7 sage: E1.is_isogenous(E5) Traceback (most recent call last): ... ValueError: Curves have different base fields: use the field parameter.
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E1 = EllipticCurve(GF(Integer(11)**Integer(2),'a'),[Integer(2),Integer(7)]); E1 Elliptic Curve defined by y^2 = x^3 + 2*x + 7 over Finite Field in a of size 11^2 >>> E1.is_isogenous(Integer(5)) Traceback (most recent call last): ... ValueError: Second argument is not an Elliptic Curve. >>> E1.is_isogenous(E1) True >>> # needs sage.rings.finite_rings >>> E2 = EllipticCurve(GF(Integer(7)**Integer(3),'b'),[Integer(3),Integer(1)]); E2 Elliptic Curve defined by y^2 = x^3 + 3*x + 1 over Finite Field in b of size 7^3 >>> E1.is_isogenous(E2) Traceback (most recent call last): ... ValueError: The base fields must have the same characteristic. >>> # needs sage.rings.finite_rings >>> E3 = EllipticCurve(GF(Integer(11)**Integer(2),'c'),[Integer(4),Integer(3)]); E3 Elliptic Curve defined by y^2 = x^3 + 4*x + 3 over Finite Field in c of size 11^2 >>> E1.is_isogenous(E3) False >>> # needs sage.rings.finite_rings >>> E4 = EllipticCurve(GF(Integer(11)**Integer(6),'d'),[Integer(6),Integer(5)]); E4 Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in d of size 11^6 >>> E1.is_isogenous(E4) True >>> # needs sage.rings.finite_rings >>> E5 = EllipticCurve(GF(Integer(11)**Integer(7),'e'),[Integer(4),Integer(2)]); E5 Elliptic Curve defined by y^2 = x^3 + 4*x + 2 over Finite Field in e of size 11^7 >>> E1.is_isogenous(E5) Traceback (most recent call last): ... ValueError: Curves have different base fields: use the field parameter.
When the field is given:
sage: # needs sage.rings.finite_rings sage: E1 = EllipticCurve(GF(13^2,'a'),[2,7]); E1 Elliptic Curve defined by y^2 = x^3 + 2*x + 7 over Finite Field in a of size 13^2 sage: E1.is_isogenous(5,GF(13^6,'f')) Traceback (most recent call last): ... ValueError: Second argument is not an Elliptic Curve. sage: E6 = EllipticCurve(GF(11^3,'g'),[9,3]); E6 Elliptic Curve defined by y^2 = x^3 + 9*x + 3 over Finite Field in g of size 11^3 sage: E1.is_isogenous(E6,QQ) Traceback (most recent call last): ... ValueError: The base fields must have the same characteristic. sage: E7 = EllipticCurve(GF(13^5,'h'),[2,9]); E7 Elliptic Curve defined by y^2 = x^3 + 2*x + 9 over Finite Field in h of size 13^5 sage: E1.is_isogenous(E7,GF(13^4,'i')) Traceback (most recent call last): ... ValueError: Field must be an extension of the base fields of both curves sage: E1.is_isogenous(E7,GF(13^10,'j')) False sage: E1.is_isogenous(E7,GF(13^30,'j')) False
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E1 = EllipticCurve(GF(Integer(13)**Integer(2),'a'),[Integer(2),Integer(7)]); E1 Elliptic Curve defined by y^2 = x^3 + 2*x + 7 over Finite Field in a of size 13^2 >>> E1.is_isogenous(Integer(5),GF(Integer(13)**Integer(6),'f')) Traceback (most recent call last): ... ValueError: Second argument is not an Elliptic Curve. >>> E6 = EllipticCurve(GF(Integer(11)**Integer(3),'g'),[Integer(9),Integer(3)]); E6 Elliptic Curve defined by y^2 = x^3 + 9*x + 3 over Finite Field in g of size 11^3 >>> E1.is_isogenous(E6,QQ) Traceback (most recent call last): ... ValueError: The base fields must have the same characteristic. >>> E7 = EllipticCurve(GF(Integer(13)**Integer(5),'h'),[Integer(2),Integer(9)]); E7 Elliptic Curve defined by y^2 = x^3 + 2*x + 9 over Finite Field in h of size 13^5 >>> E1.is_isogenous(E7,GF(Integer(13)**Integer(4),'i')) Traceback (most recent call last): ... ValueError: Field must be an extension of the base fields of both curves >>> E1.is_isogenous(E7,GF(Integer(13)**Integer(10),'j')) False >>> E1.is_isogenous(E7,GF(Integer(13)**Integer(30),'j')) False
- is_ordinary(proof=True)[source]¶
Return
Trueif this elliptic curve is ordinary, elseFalse.INPUT:
proof– boolean (default:True); ifTrue, returns a proved result. IfFalse, then a return value ofTrueis certain but a return value ofFalsemay be based on a probabilistic test. See the documentation of the functionis_j_supersingular()for more details.
EXAMPLES:
sage: F = GF(101) sage: EllipticCurve(j=F(0)).is_ordinary() False sage: EllipticCurve(j=F(1728)).is_ordinary() True sage: EllipticCurve(j=F(66)).is_ordinary() False sage: EllipticCurve(j=F(99)).is_ordinary() True
>>> from sage.all import * >>> F = GF(Integer(101)) >>> EllipticCurve(j=F(Integer(0))).is_ordinary() False >>> EllipticCurve(j=F(Integer(1728))).is_ordinary() True >>> EllipticCurve(j=F(Integer(66))).is_ordinary() False >>> EllipticCurve(j=F(Integer(99))).is_ordinary() True
- is_supersingular(proof=True)[source]¶
Return
Trueif this elliptic curve is supersingular, elseFalse.INPUT:
proof– boolean (default:True); ifTrue, returns a proved result. IfFalse, then a return value ofFalseis certain but a return value ofTruemay be based on a probabilistic test. See the documentation of the functionis_j_supersingular()for more details.
EXAMPLES:
sage: F = GF(101) sage: EllipticCurve(j=F(0)).is_supersingular() True sage: EllipticCurve(j=F(1728)).is_supersingular() False sage: EllipticCurve(j=F(66)).is_supersingular() True sage: EllipticCurve(j=F(99)).is_supersingular() False
>>> from sage.all import * >>> F = GF(Integer(101)) >>> EllipticCurve(j=F(Integer(0))).is_supersingular() True >>> EllipticCurve(j=F(Integer(1728))).is_supersingular() False >>> EllipticCurve(j=F(Integer(66))).is_supersingular() True >>> EllipticCurve(j=F(Integer(99))).is_supersingular() False
- multiplication_by_p_isogeny()[source]¶
Return the multiplication-by-\(p\) isogeny.
EXAMPLES:
sage: p = 23 sage: K.<a> = GF(p^3) sage: E = EllipticCurve(j=K.random_element()) sage: phi = E.multiplication_by_p_isogeny() sage: assert phi.degree() == p**2 sage: P = E.random_element() sage: assert phi(P) == P * p
>>> from sage.all import * >>> p = Integer(23) >>> K = GF(p**Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> E = EllipticCurve(j=K.random_element()) >>> phi = E.multiplication_by_p_isogeny() >>> assert phi.degree() == p**Integer(2) >>> P = E.random_element() >>> assert phi(P) == P * p
- order(algorithm=None, extension_degree=1)[source]¶
Return the number of points on this elliptic curve.
INPUT:
algorithm– (optional) string:'pari'– use the PARI C-library functionellcard'bsgs'– use the baby-step giant-step method asimplemented in Sage, with the Cremona-Sutherland version of Mestre’s trick
'exhaustive'– naive point counting'subfield'– reduce to a smaller field, provided that the j-invariant lies in a subfield'all'– compute cardinality with both'pari'and'bsgs'; return result if they agree or raise aAssertionErrorif they do not
extension_degree– integer \(d\) (default: 1); if the base field is \(\GF{q}\), return the cardinality ofselfover the extension \(\GF{q^d}\) of degree \(d\)
OUTPUT:
The order of the group of rational points of
selfover its base field, or over an extension field of degree \(d\) as above. The result is cached.EXAMPLES:
sage: # needs sage.rings.finite_rings sage: EllipticCurve(GF(4, 'a'), [1,2,3,4,5]).cardinality() 8 sage: k.<a> = GF(3^3) sage: l = [a^2 + 1, 2*a^2 + 2*a + 1, a^2 + a + 1, 2, 2*a] sage: EllipticCurve(k,l).cardinality() 29
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> EllipticCurve(GF(Integer(4), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 8 >>> k = GF(Integer(3)**Integer(3), names=('a',)); (a,) = k._first_ngens(1) >>> l = [a**Integer(2) + Integer(1), Integer(2)*a**Integer(2) + Integer(2)*a + Integer(1), a**Integer(2) + a + Integer(1), Integer(2), Integer(2)*a] >>> EllipticCurve(k,l).cardinality() 29
sage: # needs sage.rings.finite_rings sage: l = [1, 1, 0, 2, 0] sage: EllipticCurve(k, l).cardinality() 38
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> l = [Integer(1), Integer(1), Integer(0), Integer(2), Integer(0)] >>> EllipticCurve(k, l).cardinality() 38
An even bigger extension (which we check against Magma):
sage: # needs sage.rings.finite_rings sage: EllipticCurve(GF(3^100, 'a'), [1,2,3,4,5]).cardinality() 515377520732011331036459693969645888996929981504 sage: magma.eval("Order(EllipticCurve([GF(3^100)|1,2,3,4,5]))") # optional - magma '515377520732011331036459693969645888996929981504'
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> EllipticCurve(GF(Integer(3)**Integer(100), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 515377520732011331036459693969645888996929981504 >>> magma.eval("Order(EllipticCurve([GF(3^100)|1,2,3,4,5]))") # optional - magma '515377520732011331036459693969645888996929981504'
sage: EllipticCurve(GF(10007), [1,2,3,4,5]).cardinality() 10076 sage: EllipticCurve(GF(10007), [1,2,3,4,5]).cardinality(algorithm='pari') 10076 sage: EllipticCurve(GF(next_prime(10**20)), [1,2,3,4,5]).cardinality() 100000000011093199520
>>> from sage.all import * >>> EllipticCurve(GF(Integer(10007)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 10076 >>> EllipticCurve(GF(Integer(10007)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality(algorithm='pari') 10076 >>> EllipticCurve(GF(next_prime(Integer(10)**Integer(20))), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).cardinality() 100000000011093199520
The cardinality is cached:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(3^100, 'a'), [1,2,3,4,5]) sage: E.cardinality() is E.cardinality() True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(3)**Integer(100), 'a'), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.cardinality() is E.cardinality() True
The following is very fast since the curve is actually defined over the prime field:
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(11^100) sage: E1 = EllipticCurve(k, [3,3]) sage: N1 = E1.cardinality(algorithm='subfield'); N1 137806123398222701841183371720896367762643312000384671846835266941791510341065565176497846502742959856128 sage: E1.cardinality_pari() == N1 True sage: E2 = E1.quadratic_twist() sage: N2 = E2.cardinality(algorithm='subfield'); N2 137806123398222701841183371720896367762643312000384656816094284101308193849980588362304472492174093035876 sage: E2.cardinality_pari() == N2 True sage: N1 + N2 == 2*(k.cardinality() + 1) True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(11)**Integer(100), names=('a',)); (a,) = k._first_ngens(1) >>> E1 = EllipticCurve(k, [Integer(3),Integer(3)]) >>> N1 = E1.cardinality(algorithm='subfield'); N1 137806123398222701841183371720896367762643312000384671846835266941791510341065565176497846502742959856128 >>> E1.cardinality_pari() == N1 True >>> E2 = E1.quadratic_twist() >>> N2 = E2.cardinality(algorithm='subfield'); N2 137806123398222701841183371720896367762643312000384656816094284101308193849980588362304472492174093035876 >>> E2.cardinality_pari() == N2 True >>> N1 + N2 == Integer(2)*(k.cardinality() + Integer(1)) True
We can count points over curves defined as a reduction:
sage: # needs sage.rings.number_field sage: x = polygen(QQ) sage: K.<w> = NumberField(x^2 + x + 1) sage: EK = EllipticCurve(K, [0, 0, w, 2, 1]) sage: E = EK.base_extend(K.residue_field(2)) sage: E Elliptic Curve defined by y^2 + wbar*y = x^3 + 1 over Residue field in wbar of Fractional ideal (2) sage: E.cardinality() 7 sage: E = EK.base_extend(K.residue_field(w - 1)) sage: E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 2*x + 1 over Residue field of Fractional ideal (w - 1)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> x = polygen(QQ) >>> K = NumberField(x**Integer(2) + x + Integer(1), names=('w',)); (w,) = K._first_ngens(1) >>> EK = EllipticCurve(K, [Integer(0), Integer(0), w, Integer(2), Integer(1)]) >>> E = EK.base_extend(K.residue_field(Integer(2))) >>> E Elliptic Curve defined by y^2 + wbar*y = x^3 + 1 over Residue field in wbar of Fractional ideal (2) >>> E.cardinality() 7 >>> E = EK.base_extend(K.residue_field(w - Integer(1))) >>> E.abelian_group() Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 2*x + 1 over Residue field of Fractional ideal (w - 1)
sage: R.<x> = GF(17)[] sage: pol = R.irreducible_element(5) sage: k.<a> = R.residue_field(pol) sage: E = EllipticCurve(R, [1, x]).base_extend(k) sage: E Elliptic Curve defined by y^2 = x^3 + x + a over Residue field in a of Principal ideal (x^5 + x + 14) of Univariate Polynomial Ring in x over Finite Field of size 17 sage: E.cardinality() 1421004
>>> from sage.all import * >>> R = GF(Integer(17))['x']; (x,) = R._first_ngens(1) >>> pol = R.irreducible_element(Integer(5)) >>> k = R.residue_field(pol, names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(R, [Integer(1), x]).base_extend(k) >>> E Elliptic Curve defined by y^2 = x^3 + x + a over Residue field in a of Principal ideal (x^5 + x + 14) of Univariate Polynomial Ring in x over Finite Field of size 17 >>> E.cardinality() 1421004
- plot(*args, **kwds)[source]¶
Draw a graph of this elliptic curve over a prime finite field.
INPUT:
*args,**kwds– all other options are passed to the circle graphing primitive
EXAMPLES:
sage: E = EllipticCurve(FiniteField(17), [0,1]) sage: P = plot(E, rgbcolor=(0,0,1)) # needs sage.plot
>>> from sage.all import * >>> E = EllipticCurve(FiniteField(Integer(17)), [Integer(0),Integer(1)]) >>> P = plot(E, rgbcolor=(Integer(0),Integer(0),Integer(1))) # needs sage.plot
- points()[source]¶
Return all rational points on this elliptic curve. The list of points is cached so subsequent calls are free.
EXAMPLES:
sage: p = 5 sage: F = GF(p) sage: E = EllipticCurve(F, [1, 3]) sage: len(E.points()) 4 sage: E.order() 4 sage: E.points() [(0 : 1 : 0), (1 : 0 : 1), (4 : 1 : 1), (4 : 4 : 1)]
>>> from sage.all import * >>> p = Integer(5) >>> F = GF(p) >>> E = EllipticCurve(F, [Integer(1), Integer(3)]) >>> len(E.points()) 4 >>> E.order() 4 >>> E.points() [(0 : 1 : 0), (1 : 0 : 1), (4 : 1 : 1), (4 : 4 : 1)]
sage: K = GF((p, 2), 'a') sage: E = E.change_ring(K) sage: len(E.points()) 32 sage: E.order() 32 sage: w = E.points(); w [(0 : 1 : 0), (0 : 2*a + 4 : 1), (0 : 3*a + 1 : 1), (1 : 0 : 1), (2 : 2*a + 4 : 1), (2 : 3*a + 1 : 1), (3 : 2*a + 4 : 1), (3 : 3*a + 1 : 1), (4 : 1 : 1), (4 : 4 : 1), (a : 1 : 1), (a : 4 : 1), (a + 2 : a + 1 : 1), (a + 2 : 4*a + 4 : 1), (a + 3 : a : 1), (a + 3 : 4*a : 1), (a + 4 : 0 : 1), (2*a : 2*a : 1), (2*a : 3*a : 1), (2*a + 4 : a + 1 : 1), (2*a + 4 : 4*a + 4 : 1), (3*a + 1 : a + 3 : 1), (3*a + 1 : 4*a + 2 : 1), (3*a + 2 : 2*a + 3 : 1), (3*a + 2 : 3*a + 2 : 1), (4*a : 0 : 1), (4*a + 1 : 1 : 1), (4*a + 1 : 4 : 1), (4*a + 3 : a + 3 : 1), (4*a + 3 : 4*a + 2 : 1), (4*a + 4 : a + 4 : 1), (4*a + 4 : 4*a + 1 : 1)]
>>> from sage.all import * >>> K = GF((p, Integer(2)), 'a') >>> E = E.change_ring(K) >>> len(E.points()) 32 >>> E.order() 32 >>> w = E.points(); w [(0 : 1 : 0), (0 : 2*a + 4 : 1), (0 : 3*a + 1 : 1), (1 : 0 : 1), (2 : 2*a + 4 : 1), (2 : 3*a + 1 : 1), (3 : 2*a + 4 : 1), (3 : 3*a + 1 : 1), (4 : 1 : 1), (4 : 4 : 1), (a : 1 : 1), (a : 4 : 1), (a + 2 : a + 1 : 1), (a + 2 : 4*a + 4 : 1), (a + 3 : a : 1), (a + 3 : 4*a : 1), (a + 4 : 0 : 1), (2*a : 2*a : 1), (2*a : 3*a : 1), (2*a + 4 : a + 1 : 1), (2*a + 4 : 4*a + 4 : 1), (3*a + 1 : a + 3 : 1), (3*a + 1 : 4*a + 2 : 1), (3*a + 2 : 2*a + 3 : 1), (3*a + 2 : 3*a + 2 : 1), (4*a : 0 : 1), (4*a + 1 : 1 : 1), (4*a + 1 : 4 : 1), (4*a + 3 : a + 3 : 1), (4*a + 3 : 4*a + 2 : 1), (4*a + 4 : a + 4 : 1), (4*a + 4 : 4*a + 1 : 1)]
Note that the returned list is an immutable sorted Sequence:
sage: w[0] = 9 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
>>> from sage.all import * >>> w[Integer(0)] = Integer(9) Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
- random_element()[source]¶
Return a random point on this elliptic curve, uniformly chosen among all rational points.
ALGORITHM:
Choose the point at infinity with probability \(1/(2q + 1)\). Otherwise, take a random element from the field as x-coordinate and compute the possible y-coordinates. Return the i-th possible y-coordinate, where i is randomly chosen to be 0 or 1. If the i-th y-coordinate does not exist (either there is no point with the given x-coordinate or we hit a 2-torsion point with i == 1), try again.
This gives a uniform distribution because you can imagine \(2q + 1\) buckets, one for the point at infinity and 2 for each element of the field (representing the x-coordinates). This gives a 1-to-1 map of elliptic curve points into buckets. At every iteration, we simply choose a random bucket until we find a bucket containing a point.
AUTHORS:
Jeroen Demeyer (2014-09-09): choose points uniformly random, see Issue #16951.
EXAMPLES:
sage: k = GF(next_prime(7^5)) sage: E = EllipticCurve(k,[2,4]) sage: P = E.random_element(); P # random (16740 : 12486 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> k = GF(next_prime(Integer(7)**Integer(5))) >>> E = EllipticCurve(k,[Integer(2),Integer(4)]) >>> P = E.random_element(); P # random (16740 : 12486 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(7^5) sage: E = EllipticCurve(k,[2,4]) sage: P = E.random_element(); P # random (5*a^4 + 3*a^3 + 2*a^2 + a + 4 : 2*a^4 + 3*a^3 + 4*a^2 + a + 5 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(7)**Integer(5), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(k,[Integer(2),Integer(4)]) >>> P = E.random_element(); P # random (5*a^4 + 3*a^3 + 2*a^2 + a + 4 : 2*a^4 + 3*a^3 + 4*a^2 + a + 5 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(2^5) sage: E = EllipticCurve(k,[a^2,a,1,a+1,1]) sage: P = E.random_element(); P # random (a^4 + a : a^4 + a^3 + a^2 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(2)**Integer(5), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(k,[a**Integer(2),a,Integer(1),a+Integer(1),Integer(1)]) >>> P = E.random_element(); P # random (a^4 + a : a^4 + a^3 + a^2 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
Ensure that the entire point set is reachable:
sage: E = EllipticCurve(GF(11), [2,1]) sage: S = set() sage: while len(S) < E.cardinality(): ....: S.add(E.random_element())
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)), [Integer(2),Integer(1)]) >>> S = set() >>> while len(S) < E.cardinality(): ... S.add(E.random_element())
- random_point()[source]¶
Return a random point on this elliptic curve, uniformly chosen among all rational points.
ALGORITHM:
Choose the point at infinity with probability \(1/(2q + 1)\). Otherwise, take a random element from the field as x-coordinate and compute the possible y-coordinates. Return the i-th possible y-coordinate, where i is randomly chosen to be 0 or 1. If the i-th y-coordinate does not exist (either there is no point with the given x-coordinate or we hit a 2-torsion point with i == 1), try again.
This gives a uniform distribution because you can imagine \(2q + 1\) buckets, one for the point at infinity and 2 for each element of the field (representing the x-coordinates). This gives a 1-to-1 map of elliptic curve points into buckets. At every iteration, we simply choose a random bucket until we find a bucket containing a point.
AUTHORS:
Jeroen Demeyer (2014-09-09): choose points uniformly random, see Issue #16951.
EXAMPLES:
sage: k = GF(next_prime(7^5)) sage: E = EllipticCurve(k,[2,4]) sage: P = E.random_element(); P # random (16740 : 12486 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> k = GF(next_prime(Integer(7)**Integer(5))) >>> E = EllipticCurve(k,[Integer(2),Integer(4)]) >>> P = E.random_element(); P # random (16740 : 12486 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(7^5) sage: E = EllipticCurve(k,[2,4]) sage: P = E.random_element(); P # random (5*a^4 + 3*a^3 + 2*a^2 + a + 4 : 2*a^4 + 3*a^3 + 4*a^2 + a + 5 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(7)**Integer(5), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(k,[Integer(2),Integer(4)]) >>> P = E.random_element(); P # random (5*a^4 + 3*a^3 + 2*a^2 + a + 4 : 2*a^4 + 3*a^3 + 4*a^2 + a + 5 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
sage: # needs sage.rings.finite_rings sage: k.<a> = GF(2^5) sage: E = EllipticCurve(k,[a^2,a,1,a+1,1]) sage: P = E.random_element(); P # random (a^4 + a : a^4 + a^3 + a^2 : 1) sage: type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> sage: P in E True
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = GF(Integer(2)**Integer(5), names=('a',)); (a,) = k._first_ngens(1) >>> E = EllipticCurve(k,[a**Integer(2),a,Integer(1),a+Integer(1),Integer(1)]) >>> P = E.random_element(); P # random (a^4 + a : a^4 + a^3 + a^2 : 1) >>> type(P) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field'> >>> P in E True
Ensure that the entire point set is reachable:
sage: E = EllipticCurve(GF(11), [2,1]) sage: S = set() sage: while len(S) < E.cardinality(): ....: S.add(E.random_element())
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(11)), [Integer(2),Integer(1)]) >>> S = set() >>> while len(S) < E.cardinality(): ... S.add(E.random_element())
- rational_points()[source]¶
Return all rational points on this elliptic curve. The list of points is cached so subsequent calls are free.
EXAMPLES:
sage: p = 5 sage: F = GF(p) sage: E = EllipticCurve(F, [1, 3]) sage: len(E.points()) 4 sage: E.order() 4 sage: E.points() [(0 : 1 : 0), (1 : 0 : 1), (4 : 1 : 1), (4 : 4 : 1)]
>>> from sage.all import * >>> p = Integer(5) >>> F = GF(p) >>> E = EllipticCurve(F, [Integer(1), Integer(3)]) >>> len(E.points()) 4 >>> E.order() 4 >>> E.points() [(0 : 1 : 0), (1 : 0 : 1), (4 : 1 : 1), (4 : 4 : 1)]
sage: K = GF((p, 2), 'a') sage: E = E.change_ring(K) sage: len(E.points()) 32 sage: E.order() 32 sage: w = E.points(); w [(0 : 1 : 0), (0 : 2*a + 4 : 1), (0 : 3*a + 1 : 1), (1 : 0 : 1), (2 : 2*a + 4 : 1), (2 : 3*a + 1 : 1), (3 : 2*a + 4 : 1), (3 : 3*a + 1 : 1), (4 : 1 : 1), (4 : 4 : 1), (a : 1 : 1), (a : 4 : 1), (a + 2 : a + 1 : 1), (a + 2 : 4*a + 4 : 1), (a + 3 : a : 1), (a + 3 : 4*a : 1), (a + 4 : 0 : 1), (2*a : 2*a : 1), (2*a : 3*a : 1), (2*a + 4 : a + 1 : 1), (2*a + 4 : 4*a + 4 : 1), (3*a + 1 : a + 3 : 1), (3*a + 1 : 4*a + 2 : 1), (3*a + 2 : 2*a + 3 : 1), (3*a + 2 : 3*a + 2 : 1), (4*a : 0 : 1), (4*a + 1 : 1 : 1), (4*a + 1 : 4 : 1), (4*a + 3 : a + 3 : 1), (4*a + 3 : 4*a + 2 : 1), (4*a + 4 : a + 4 : 1), (4*a + 4 : 4*a + 1 : 1)]
>>> from sage.all import * >>> K = GF((p, Integer(2)), 'a') >>> E = E.change_ring(K) >>> len(E.points()) 32 >>> E.order() 32 >>> w = E.points(); w [(0 : 1 : 0), (0 : 2*a + 4 : 1), (0 : 3*a + 1 : 1), (1 : 0 : 1), (2 : 2*a + 4 : 1), (2 : 3*a + 1 : 1), (3 : 2*a + 4 : 1), (3 : 3*a + 1 : 1), (4 : 1 : 1), (4 : 4 : 1), (a : 1 : 1), (a : 4 : 1), (a + 2 : a + 1 : 1), (a + 2 : 4*a + 4 : 1), (a + 3 : a : 1), (a + 3 : 4*a : 1), (a + 4 : 0 : 1), (2*a : 2*a : 1), (2*a : 3*a : 1), (2*a + 4 : a + 1 : 1), (2*a + 4 : 4*a + 4 : 1), (3*a + 1 : a + 3 : 1), (3*a + 1 : 4*a + 2 : 1), (3*a + 2 : 2*a + 3 : 1), (3*a + 2 : 3*a + 2 : 1), (4*a : 0 : 1), (4*a + 1 : 1 : 1), (4*a + 1 : 4 : 1), (4*a + 3 : a + 3 : 1), (4*a + 3 : 4*a + 2 : 1), (4*a + 4 : a + 4 : 1), (4*a + 4 : 4*a + 1 : 1)]
Note that the returned list is an immutable sorted Sequence:
sage: w[0] = 9 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
>>> from sage.all import * >>> w[Integer(0)] = Integer(9) Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
- set_order(value, check, num_checks)[source]¶
Set the value of
self._ordertovalue.Use this when you know a priori the order of the curve to avoid a potentially expensive order calculation.
INPUT:
value– integer in the Hasse-Weil range for this curvecheck– boolean (default:True); whether or not to run sanity checks on the inputnum_checks– integer (default: \(8\)); ifcheckisTrue, the number of times to check whethervaluetimes a random point on this curve equals the identity
OUTPUT: none
EXAMPLES:
This example illustrates basic usage:
sage: E = EllipticCurve(GF(7), [0, 1]) # This curve has order 12 sage: E.set_order(12) sage: E.order() 12 sage: E.order() * E.random_point() (0 : 1 : 0)
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(7)), [Integer(0), Integer(1)]) # This curve has order 12 >>> E.set_order(Integer(12)) >>> E.order() 12 >>> E.order() * E.random_point() (0 : 1 : 0)
We now give a more interesting case, the NIST-P521 curve. Its order is too big to calculate with Sage, and takes a long time using other packages, so it is very useful here:
sage: p = 2^521 - 1 sage: prev_proof_state = proof.arithmetic() sage: proof.arithmetic(False) # turn off primality checking sage: F = GF(p) sage: A = p - 3 sage: B = 1093849038073734274511112390766805569936207598951683748994586394495953116150735016013708737573759623248592132296706313309438452531591012912142327488478985984 sage: q = 6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449 sage: E = EllipticCurve([F(A), F(B)]) sage: E.set_order(q) sage: G = E.random_point() sage: G.order() * G # This takes practically no time. (0 : 1 : 0) sage: proof.arithmetic(prev_proof_state) # restore state
>>> from sage.all import * >>> p = Integer(2)**Integer(521) - Integer(1) >>> prev_proof_state = proof.arithmetic() >>> proof.arithmetic(False) # turn off primality checking >>> F = GF(p) >>> A = p - Integer(3) >>> B = Integer(1093849038073734274511112390766805569936207598951683748994586394495953116150735016013708737573759623248592132296706313309438452531591012912142327488478985984) >>> q = Integer(6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449) >>> E = EllipticCurve([F(A), F(B)]) >>> E.set_order(q) >>> G = E.random_point() >>> G.order() * G # This takes practically no time. (0 : 1 : 0) >>> proof.arithmetic(prev_proof_state) # restore state
It is an error to pass a value which is not an integer in the Hasse-Weil range:
sage: E = EllipticCurve(GF(7), [0, 1]) # This curve has order 12 sage: E.set_order("hi") Traceback (most recent call last): ... TypeError: unable to convert 'hi' to an integer sage: E.set_order(0) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 0 sage: E.set_order(1000) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 1000
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(7)), [Integer(0), Integer(1)]) # This curve has order 12 >>> E.set_order("hi") Traceback (most recent call last): ... TypeError: unable to convert 'hi' to an integer >>> E.set_order(Integer(0)) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 0 >>> E.set_order(Integer(1000)) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 1000
It is also very likely an error to pass a value which is not the actual order of this curve. How unlikely is determined by
num_checks, the factorization of the actual order, and the actual group structure:sage: E = EllipticCurve(GF(1009), [0, 1]) # This curve has order 948 sage: E.set_order(947) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 1009 does not have order 947
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(1009)), [Integer(0), Integer(1)]) # This curve has order 948 >>> E.set_order(Integer(947)) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 1009 does not have order 947
For curves over small finite fields, the order is cheap to compute, so it is computed directly and compared:
sage: E = EllipticCurve(GF(7), [0, 1]) # This curve has order 12 sage: E.set_order(11) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 11
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(7)), [Integer(0), Integer(1)]) # This curve has order 12 >>> E.set_order(Integer(11)) Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 does not have order 11
Todo
Add provable correctness check by computing the abelian group structure and comparing.
AUTHORS:
Mariah Lenox (2011-02-16): Initial implementation
Gareth Ma (2024-01-21): Fix bug for small curves
- torsion_basis(n)[source]¶
Return a basis of the \(n\)-torsion subgroup of this elliptic curve, assuming it is fully rational.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(62207^2), [1,0]) sage: E.abelian_group() Additive abelian group isomorphic to Z/62208 + Z/62208 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 62207^2 sage: PA,QA = E.torsion_basis(2^8) sage: PA.weil_pairing(QA, 2^8).multiplicative_order() 256 sage: PB,QB = E.torsion_basis(3^5) sage: PB.weil_pairing(QB, 3^5).multiplicative_order() 243
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(62207)**Integer(2)), [Integer(1),Integer(0)]) >>> E.abelian_group() Additive abelian group isomorphic to Z/62208 + Z/62208 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 62207^2 >>> PA,QA = E.torsion_basis(Integer(2)**Integer(8)) >>> PA.weil_pairing(QA, Integer(2)**Integer(8)).multiplicative_order() 256 >>> PB,QB = E.torsion_basis(Integer(3)**Integer(5)) >>> PB.weil_pairing(QB, Integer(3)**Integer(5)).multiplicative_order() 243
sage: E = EllipticCurve(GF(101), [4,4]) sage: E.torsion_basis(23) Traceback (most recent call last): ... ValueError: curve does not have full rational 23-torsion sage: F = E.division_field(23); F Finite Field in t of size 101^11 sage: EE = E.change_ring(F) sage: P, Q = EE.torsion_basis(23) sage: P # random (89*z11^10 + 51*z11^9 + 96*z11^8 + 8*z11^7 + 67*z11^6 + 31*z11^5 + 55*z11^4 + 59*z11^3 + 28*z11^2 + 8*z11 + 88 : 40*z11^10 + 33*z11^9 + 80*z11^8 + 87*z11^7 + 97*z11^6 + 69*z11^5 + 56*z11^4 + 17*z11^3 + 26*z11^2 + 69*z11 + 11 : 1) sage: Q # random (25*z11^10 + 61*z11^9 + 49*z11^8 + 17*z11^7 + 80*z11^6 + 20*z11^5 + 49*z11^4 + 52*z11^3 + 61*z11^2 + 27*z11 + 61 : 60*z11^10 + 91*z11^9 + 89*z11^8 + 7*z11^7 + 63*z11^6 + 55*z11^5 + 23*z11^4 + 17*z11^3 + 90*z11^2 + 91*z11 + 68 : 1)
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(101)), [Integer(4),Integer(4)]) >>> E.torsion_basis(Integer(23)) Traceback (most recent call last): ... ValueError: curve does not have full rational 23-torsion >>> F = E.division_field(Integer(23)); F Finite Field in t of size 101^11 >>> EE = E.change_ring(F) >>> P, Q = EE.torsion_basis(Integer(23)) >>> P # random (89*z11^10 + 51*z11^9 + 96*z11^8 + 8*z11^7 + 67*z11^6 + 31*z11^5 + 55*z11^4 + 59*z11^3 + 28*z11^2 + 8*z11 + 88 : 40*z11^10 + 33*z11^9 + 80*z11^8 + 87*z11^7 + 97*z11^6 + 69*z11^5 + 56*z11^4 + 17*z11^3 + 26*z11^2 + 69*z11 + 11 : 1) >>> Q # random (25*z11^10 + 61*z11^9 + 49*z11^8 + 17*z11^7 + 80*z11^6 + 20*z11^5 + 49*z11^4 + 52*z11^3 + 61*z11^2 + 27*z11 + 61 : 60*z11^10 + 91*z11^9 + 89*z11^8 + 7*z11^7 + 63*z11^6 + 55*z11^5 + 23*z11^4 + 17*z11^3 + 90*z11^2 + 91*z11 + 68 : 1)
See also
Use
division_field()to determine a field extension containing the full \(\ell\)-torsion subgroup.ALGORITHM:
This method currently uses
abelian_group()andAdditiveAbelianGroupWrapper.torsion_subgroup().
- trace_of_frobenius()[source]¶
Return the trace of Frobenius acting on this elliptic curve.
Note
This computes the curve cardinality, which may be time-consuming.
EXAMPLES:
sage: E = EllipticCurve(GF(101),[2,3]) sage: E.trace_of_frobenius() 6 sage: E = EllipticCurve(GF(11^5,'a'),[2,5]) # needs sage.rings.finite_rings sage: E.trace_of_frobenius() # needs sage.rings.finite_rings 802
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(101)),[Integer(2),Integer(3)]) >>> E.trace_of_frobenius() 6 >>> E = EllipticCurve(GF(Integer(11)**Integer(5),'a'),[Integer(2),Integer(5)]) # needs sage.rings.finite_rings >>> E.trace_of_frobenius() # needs sage.rings.finite_rings 802
The following shows that the issue from Issue #2849 is fixed:
sage: E = EllipticCurve(GF(3^5,'a'),[-1,-1]) # needs sage.rings.finite_rings sage: E.trace_of_frobenius() # needs sage.rings.finite_rings -27
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(3)**Integer(5),'a'),[-Integer(1),-Integer(1)]) # needs sage.rings.finite_rings >>> E.trace_of_frobenius() # needs sage.rings.finite_rings -27
- twists()[source]¶
Return a list of \(k\)-isomorphism representatives of all twists of this elliptic curve, where \(k\) is the base field.
The input curve appears as the first entry of the result.
Note
A twist of \(E/k\) is an elliptic curve \(E'\) defined over \(k\) that is isomorphic to \(E\) over the algebraic closure \(\bar k\).
Most elliptic curves over a finite field only admit a single nontrivial twist (the quadratic twist); the only exceptions are curves with \(j\)-invariant \(0\) or \(1728\).
In all cases the sum over all the twists \(E'\) of \(1/|Aut(E')|\) is 1.
EXAMPLES:
sage: E = EllipticCurve(GF(97), [1,1]) sage: E.j_invariant() 54 sage: E.twists() [Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(97)), [Integer(1),Integer(1)]) >>> E.j_invariant() 54 >>> E.twists() [Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
sage: E = EllipticCurve(GF(97), [1,0]) sage: E.j_invariant() 79 sage: E.twists() [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(97)), [Integer(1),Integer(0)]) >>> E.j_invariant() 79 >>> E.twists() [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
sage: E = EllipticCurve(GF(97), [0,1]) sage: E.j_invariant() 0 sage: E.twists() [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(97)), [Integer(0),Integer(1)]) >>> E.j_invariant() 0 >>> E.twists() [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97, Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 97]
This can be useful to quickly compute a list of all elliptic curves over a finite field \(k\) up to \(k\)-isomorphism:
sage: Es = [E for j in GF(13) for E in EllipticCurve(j=j).twists()] sage: len(Es) 32 sage: Es [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 13, ... Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 13]
>>> from sage.all import * >>> Es = [E for j in GF(Integer(13)) for E in EllipticCurve(j=j).twists()] >>> len(Es) 32 >>> Es [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 13, ... Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 13]
In characteristic 3, the number of twists is 2 except for \(j=0=1728\), when there are either 4 or 6 depending on whether the field has odd or even degree over \(\GF{3}\):
sage: # needs sage.rings.finite_rings sage: K = GF(3**5) sage: [E.ainvs() for E in EllipticCurve(j=K(1)).twists()] [(0, 1, 0, 0, 2), (0, z5, 0, 0, 2*z5^3)] sage: # needs sage.rings.finite_rings sage: K = GF(3**5) sage: [E.ainvs() for E in EllipticCurve(j=K(0)).twists()] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 0), (0, 0, 0, 2, z5^4 + z5^3 + z5^2), (0, 0, 0, 2, 2*z5^4 + 2*z5^3 + 2*z5^2)] sage: # needs sage.rings.finite_rings sage: K = GF(3**4) sage: [E.ainvs() for E in EllipticCurve(j=K(1)).twists()] [(0, 1, 0, 0, 2), (0, z4, 0, 0, 2*z4^3)] sage: # needs sage.rings.finite_rings sage: K = GF(3**4) sage: [E.ainvs() for E in EllipticCurve(j=K(0)).twists()] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, 1, 0), (0, 0, 0, 1, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, z4, 0), (0, 0, 0, z4^3, 0)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(5)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(1))).twists()] [(0, 1, 0, 0, 2), (0, z5, 0, 0, 2*z5^3)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(5)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(0))).twists()] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 0), (0, 0, 0, 2, z5^4 + z5^3 + z5^2), (0, 0, 0, 2, 2*z5^4 + 2*z5^3 + 2*z5^2)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(4)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(1))).twists()] [(0, 1, 0, 0, 2), (0, z4, 0, 0, 2*z4^3)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(4)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(0))).twists()] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, 1, 0), (0, 0, 0, 1, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, z4, 0), (0, 0, 0, z4^3, 0)]
In characteristic 2, the number of twists is 2 except for \(j=0=1728\), when there are either 3 or 7 depending on whether the field has odd or even degree over \(\GF{2}\):
sage: # needs sage.rings.finite_rings sage: K = GF(2**7) sage: [E.ainvs() for E in EllipticCurve(j=K(1)).twists()] [(1, 0, 0, 0, 1), (1, 1, 0, 0, 1)] sage: # needs sage.rings.finite_rings sage: K = GF(2**7) sage: [E.ainvs() for E in EllipticCurve(j=K(0)).twists()] [(0, 0, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 1, 1, 1)] sage: # needs sage.rings.finite_rings sage: K = GF(2**8) sage: [E.ainvs() for E in EllipticCurve(j=K(1)).twists()] # random [(1, 0, 0, 0, 1), (1, z8^7 + z8^6 + z8^5 + z8^4 + z8^2 + z8, 0, 0, 1)] sage: # needs sage.rings.finite_rings sage: K = GF(2**8) sage: [E.ainvs() for E in EllipticCurve(j=K(0)).twists()] # random [(0, 0, 1, 0, 0), (0, 0, 1, 0, z8^5 + z8^4 + z8^3), (0, 0, 1, z8^6 + z8^5 + z8^2 + 1, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, z8^3 + z8^2 + 1), (0, 0, z8^6 + z8^3 + z8^2, 0, 0), (0, 0, z8^6 + z8^3 + z8^2, 0, z8^3 + z8^2)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(7)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(1))).twists()] [(1, 0, 0, 0, 1), (1, 1, 0, 0, 1)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(7)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(0))).twists()] [(0, 0, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 1, 1, 1)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(8)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(1))).twists()] # random [(1, 0, 0, 0, 1), (1, z8^7 + z8^6 + z8^5 + z8^4 + z8^2 + z8, 0, 0, 1)] >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(8)) >>> [E.ainvs() for E in EllipticCurve(j=K(Integer(0))).twists()] # random [(0, 0, 1, 0, 0), (0, 0, 1, 0, z8^5 + z8^4 + z8^3), (0, 0, 1, z8^6 + z8^5 + z8^2 + 1, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, z8^3 + z8^2 + 1), (0, 0, z8^6 + z8^3 + z8^2, 0, 0), (0, 0, z8^6 + z8^3 + z8^2, 0, z8^3 + z8^2)]
- sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_with_order(m, D)[source]¶
Return an iterator for elliptic curves over finite fields with the given order. The curves are computed using the Complex Multiplication (CM) method.
A
Factorizationcan be passed form, in which case the algorithm is more efficient.If
Dis specified, it is used as the discriminant.EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_finite_field import EllipticCurve_with_order sage: E = next(EllipticCurve_with_order(1234)); E # random Elliptic Curve defined by y^2 = x^3 + 1142*x + 1209 over Finite Field of size 1237 sage: E.order() == 1234 True
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import EllipticCurve_with_order >>> E = next(EllipticCurve_with_order(Integer(1234))); E # random Elliptic Curve defined by y^2 = x^3 + 1142*x + 1209 over Finite Field of size 1237 >>> E.order() == Integer(1234) True
When
iteris set, the function returns an iterator of all elliptic curves with the given order:sage: from sage.schemes.elliptic_curves.ell_finite_field import EllipticCurve_with_order sage: it = EllipticCurve_with_order(21); it <generator object EllipticCurve_with_order at 0x...> sage: E = next(it); E # random Elliptic Curve defined by y^2 = x^3 + 6*x + 14 over Finite Field of size 23 sage: E.order() == 21 True sage: Es = [E] + list(it); Es # random [Elliptic Curve defined by y^2 = x^3 + 6*x + 14 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 12*x + 4 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 5*x + 2 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + (z2+3) over Finite Field in z2 of size 5^2, Elliptic Curve defined by y^2 = x^3 + (2*z2+2) over Finite Field in z2 of size 5^2, Elliptic Curve defined by y^2 = x^3 + 7*x + 1 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 17*x + 10 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 5*x + 12 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 9*x + 1 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 7*x + 6 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + z3^2*x^2 + (2*z3^2+z3) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 = x^3 + (z3^2+2*z3+1)*x^2 + (2*z3^2+2*z3) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 = x^3 + (z3^2+z3+1)*x^2 + (2*z3^2+1) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 + (z4^2+z4+1)*y = x^3 over Finite Field in z4 of size 2^4, Elliptic Curve defined by y^2 + (z4^2+z4)*y = x^3 over Finite Field in z4 of size 2^4, Elliptic Curve defined by y^2 = x^3 + 18*x + 26 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 11*x + 19 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 19 over Finite Field of size 31, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 13] sage: all(E.order() == 21 for E in Es) True
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import EllipticCurve_with_order >>> it = EllipticCurve_with_order(Integer(21)); it <generator object EllipticCurve_with_order at 0x...> >>> E = next(it); E # random Elliptic Curve defined by y^2 = x^3 + 6*x + 14 over Finite Field of size 23 >>> E.order() == Integer(21) True >>> Es = [E] + list(it); Es # random [Elliptic Curve defined by y^2 = x^3 + 6*x + 14 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 12*x + 4 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 5*x + 2 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + (z2+3) over Finite Field in z2 of size 5^2, Elliptic Curve defined by y^2 = x^3 + (2*z2+2) over Finite Field in z2 of size 5^2, Elliptic Curve defined by y^2 = x^3 + 7*x + 1 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 17*x + 10 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 5*x + 12 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 9*x + 1 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 7*x + 6 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + z3^2*x^2 + (2*z3^2+z3) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 = x^3 + (z3^2+2*z3+1)*x^2 + (2*z3^2+2*z3) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 = x^3 + (z3^2+z3+1)*x^2 + (2*z3^2+1) over Finite Field in z3 of size 3^3, Elliptic Curve defined by y^2 + (z4^2+z4+1)*y = x^3 over Finite Field in z4 of size 2^4, Elliptic Curve defined by y^2 + (z4^2+z4)*y = x^3 over Finite Field in z4 of size 2^4, Elliptic Curve defined by y^2 = x^3 + 18*x + 26 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 11*x + 19 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 19 over Finite Field of size 31, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 13] >>> all(E.order() == Integer(21) for E in Es) True
Indeed, we can verify that this is correct. Hasse’s bounds tell us that \(p \leq 50\) (approximately), and the rest can be checked via bruteforce:
sage: for p in prime_range(50): ....: for j in range(p): ....: E0 = EllipticCurve(GF(p), j=j) ....: for Et in E0.twists(): ....: if Et.order() == 21: ....: assert any(Et.is_isomorphic(E) for E in Es)
>>> from sage.all import * >>> for p in prime_range(Integer(50)): ... for j in range(p): ... E0 = EllipticCurve(GF(p), j=j) ... for Et in E0.twists(): ... if Et.order() == Integer(21): ... assert any(Et.is_isomorphic(E) for E in Es)
Note
The output curves are not deterministic, as
EllipticCurve_finite_field.twists()is not deterministic. However, the order of the j-invariants and base fields is fixed.AUTHORS:
Gareth Ma and Giacomo Pope (Sage Days 123): initial version
- sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_with_prime_order(N)[source]¶
Given a prime number
N, find another prime number \(p\) and construct an elliptic curve \(E\) defined over \(\mathbb F_p\) such that \(\#E(\mathbb F_p) = N\).INPUT:
N– integer; the order for which we seek an elliptic curve. Must be a prime number.
OUTPUT: an iterator of (some) elliptic curves \(E/\mathbb F_p\) of order
NALGORITHM:
Our algorithm is based on [BS2007], Algorithm 2.2, but we deviate for several key steps. Firstly, the authors in the paper perform the search for a suitable \(D\) incrementally, by enlarging the table \(S\) by \(log(N)\)-size interval of primes \(p\) and testing all products of distinct primes \(p\) (or rather \(p^*\)). We find this difficult to implement without testing duplicate \(D\)s, so we instead enlarge the table one prime at a time (effectively replacing \([r\log(N), (r + 1)\log(N)]\) in the paper by \([r, r]\)). To compensate for the speed loss, we begin the algorithm by prefilling \(S\) with the primes below \(1000\) (satisfying quadratic reciprocity properties). The constant \(1000\) is determined experimentally to be fast for many purposes, and for most \(N\) we tested we are able to find a suitable small \(D\) without increasing the size of \(S\).
The paper also doesn’t specify how to enumerate such \(D\)s, which recall should be product of distinct values in the table \(S\). We implement this with a priority queue (min heap), which also allows us to search for the suitable \(D\)s in increasing (absolute value) order. This is suitable for the algorithm because smaller \(D\) means the Hilbert class polynomial is computed quicker.
Finally, to avoid repeatedly testing the same \(D\)s, we require the latest prime to be added to the table to be included as a factor of \(D\) (see code for more explanation). As we need to find integers \(x, y\) such that \(x^2 + (-D)y^2 = 4N\) with \(D < 0\) and \(N\) prime, we actually need \(|D| \leq 4N\), so we terminate the algorithm when the primes in the table are larger than that bound. This makes the iterator return all curves it can find in finite time.
ALGORITHM: Based on [BS2007], Algorithm 2.2
EXAMPLES:
sage: N = 8314040072427107567 sage: E = next(EllipticCurve_with_prime_order(N)) sage: E Elliptic Curve defined by y^2 = x^3 + 4757897140353078952*x + 1841350074072114366 over Finite Field of size 8314040074357871443 sage: E.has_order(N) True
>>> from sage.all import * >>> N = Integer(8314040072427107567) >>> E = next(EllipticCurve_with_prime_order(N)) >>> E Elliptic Curve defined by y^2 = x^3 + 4757897140353078952*x + 1841350074072114366 over Finite Field of size 8314040074357871443 >>> E.has_order(N) True
The returned curves are sometimes random because
twists()is not deterministic. However, it’s always isomorphic:sage: E = next(EllipticCurve_with_prime_order(23)); E # random Elliptic Curve defined by y^2 = x^3 + 12*x + 6 over Finite Field of size 17 sage: E.is_isomorphic(EllipticCurve(GF(17), [3, 5])) True
>>> from sage.all import * >>> E = next(EllipticCurve_with_prime_order(Integer(23))); E # random Elliptic Curve defined by y^2 = x^3 + 12*x + 6 over Finite Field of size 17 >>> E.is_isomorphic(EllipticCurve(GF(Integer(17)), [Integer(3), Integer(5)])) True
You can directly iterate over the iterator; here only on the first 10 curves:
sage: N = 54675917 sage: for _, E in zip(range(10), EllipticCurve_with_prime_order(N)): ....: assert E.has_order(N)
>>> from sage.all import * >>> N = Integer(54675917) >>> for _, E in zip(range(Integer(10)), EllipticCurve_with_prime_order(N)): ... assert E.has_order(N)
It works for large primes:
sage: N = 2666207849820848272386538889527600954292544013630953455833 sage: E = next(EllipticCurve_with_prime_order(N)); E Elliptic Curve defined by y^2 = x^3 + 2666207849820848272386538889427721639173508298483739490459*x + 77986137112576 over Finite Field of size 2666207849820848272386538889427721639173508298487130585243 sage: E.has_order(N) True
>>> from sage.all import * >>> N = Integer(2666207849820848272386538889527600954292544013630953455833) >>> E = next(EllipticCurve_with_prime_order(N)); E Elliptic Curve defined by y^2 = x^3 + 2666207849820848272386538889427721639173508298483739490459*x + 77986137112576 over Finite Field of size 2666207849820848272386538889427721639173508298487130585243 >>> E.has_order(N) True
Another example for large primes:
sage: N = next_prime(2^256) sage: E = next(EllipticCurve_with_prime_order(N)); E # random Elliptic Curve defined by y^2 = x^3 + 6056521267553273205988520276135607487700943205131813669424576873701361709521*x + 86942739955486781674010637133214195706465136689012129911736706024465988573567 over Finite Field of size 115792089237316195423570985008687907853847329310253429036565151476471048389761 sage: E.j_invariant() 111836223967433630316209796253554285080540088646141285337487360944738698436350 sage: E.has_order(N) True
>>> from sage.all import * >>> N = next_prime(Integer(2)**Integer(256)) >>> E = next(EllipticCurve_with_prime_order(N)); E # random Elliptic Curve defined by y^2 = x^3 + 6056521267553273205988520276135607487700943205131813669424576873701361709521*x + 86942739955486781674010637133214195706465136689012129911736706024465988573567 over Finite Field of size 115792089237316195423570985008687907853847329310253429036565151476471048389761 >>> E.j_invariant() 111836223967433630316209796253554285080540088646141285337487360944738698436350 >>> E.has_order(N) True
Note that the iterator does not return all curves with the given order:
sage: any(E.base_ring() is GF(7) for E in EllipticCurve_with_prime_order(7)) False sage: EllipticCurve(GF(7), [0, 5]).order() 7
>>> from sage.all import * >>> any(E.base_ring() is GF(Integer(7)) for E in EllipticCurve_with_prime_order(Integer(7))) False >>> EllipticCurve(GF(Integer(7)), [Integer(0), Integer(5)]).order() 7
However, experimentally it returns many of them. Here it returns all of them:
sage: N = 23 sage: set_random_seed(1337) # as the function returns random twists of curves sage: curves = list(EllipticCurve_with_prime_order(N)); curves # random [Elliptic Curve defined by y^2 = x^3 + 3*x + 5 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 19*x + 14 over Finite Field of size 31, Elliptic Curve defined by y^2 = x^3 + 2*x + 9 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 7*x + 18 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 20*x + 20 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 10*x + 16 over Finite Field of size 23] sage: import itertools sage: # These are the only primes, by the Hasse-Weil bound sage: for q in prime_range(17, 35): ....: K = GF(q) ....: for u in itertools.product(range(q), repeat=2): ....: try: E = EllipticCurve(GF(q), u) ....: except ArithmeticError: continue ....: if E.has_order(N): ....: assert any(E.is_isomorphic(E_) for E_ in curves)
>>> from sage.all import * >>> N = Integer(23) >>> set_random_seed(Integer(1337)) # as the function returns random twists of curves >>> curves = list(EllipticCurve_with_prime_order(N)); curves # random [Elliptic Curve defined by y^2 = x^3 + 3*x + 5 over Finite Field of size 17, Elliptic Curve defined by y^2 = x^3 + 19*x + 14 over Finite Field of size 31, Elliptic Curve defined by y^2 = x^3 + 2*x + 9 over Finite Field of size 19, Elliptic Curve defined by y^2 = x^3 + 7*x + 18 over Finite Field of size 29, Elliptic Curve defined by y^2 = x^3 + 20*x + 20 over Finite Field of size 23, Elliptic Curve defined by y^2 = x^3 + 10*x + 16 over Finite Field of size 23] >>> import itertools >>> # These are the only primes, by the Hasse-Weil bound >>> for q in prime_range(Integer(17), Integer(35)): ... K = GF(q) ... for u in itertools.product(range(q), repeat=Integer(2)): ... try: E = EllipticCurve(GF(q), u) ... except ArithmeticError: continue ... if E.has_order(N): ... assert any(E.is_isomorphic(E_) for E_ in curves)
The algorithm is efficient for small
Ndue to the low number of suitable discriminants (see theabs_products_underinternal function of the code for details):sage: len(list(EllipticCurve_with_prime_order(next_prime(5000)))) 534 sage: len(list(EllipticCurve_with_prime_order(next_prime(50000)))) # long time (6s) 3841
>>> from sage.all import * >>> len(list(EllipticCurve_with_prime_order(next_prime(Integer(5000))))) 534 >>> len(list(EllipticCurve_with_prime_order(next_prime(Integer(50000))))) # long time (6s) 3841
There is different verbose data for level \(2\) to \(4\), though level \(3\) rarely logs anything (it logs when a new prime \(p\) is added to the smoothness bound):
sage: from sage.misc.verbose import set_verbose sage: set_random_seed(1337) # as the function returns random twists of curves sage: for _, E in zip(range(3), EllipticCurve_with_prime_order(10^9 + 7)): ....: print(E) Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307 sage: set_verbose(2) sage: set_random_seed(1337) sage: for _, E in zip(range(3), EllipticCurve_with_prime_order(10^9 + 7)): ....: print(E) verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163 Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307 sage: set_verbose(4) sage: set_random_seed(1337) sage: for _, E in zip(range(3), EllipticCurve_with_prime_order(10^9 + 7)): ....: print(E) verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-19 ... verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-163 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163 Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-179 ... verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-667 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307
>>> from sage.all import * >>> from sage.misc.verbose import set_verbose >>> set_random_seed(Integer(1337)) # as the function returns random twists of curves >>> for _, E in zip(range(Integer(3)), EllipticCurve_with_prime_order(Integer(10)**Integer(9) + Integer(7))): ... print(E) Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307 >>> set_verbose(Integer(2)) >>> set_random_seed(Integer(1337)) >>> for _, E in zip(range(Integer(3)), EllipticCurve_with_prime_order(Integer(10)**Integer(9) + Integer(7))): ... print(E) verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163 Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307 >>> set_verbose(Integer(4)) >>> set_random_seed(Integer(1337)) >>> for _, E in zip(range(Integer(3)), EllipticCurve_with_prime_order(Integer(10)**Integer(9) + Integer(7))): ... print(E) verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-19 ... verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-163 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163 Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437 verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-179 ... verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-667 verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667 Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307 Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307
AUTHORS:
Martin Grenouilloux, Gareth Ma (2024-09): initial implementation
- sage.schemes.elliptic_curves.ell_finite_field.curves_with_j_0(K)[source]¶
Return a complete list of pairwise nonisomorphic elliptic curves with \(j\)-invariant 0 over the finite field \(K\).
Note
In characteristics 2 and 3 this function simply calls
curves_with_j_0_char2orcurves_with_j_0_char3. Otherwise there are either 2 or 6 curves, parametrised by \(K^*/(K^*)^6\).Examples:
For \(K=\GF{q}\) where \(q\equiv1\mod{6}\) there are six curves, the sextic twists of \(y^2=x^3+1\):
sage: # needs sage.rings.finite_rings sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0 sage: sorted(curves_with_j_0(GF(7)), key = lambda E: E.a_invariants()) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 5 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 6 over Finite Field of size 7] sage: curves = curves_with_j_0(GF(25)); len(curves) 6 sage: all(not curves[i].is_isomorphic(curves[j]) for i in range(6) for j in range(i + 1, 6)) True sage: set(E.j_invariant() for E in curves) {0}
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0 >>> sorted(curves_with_j_0(GF(Integer(7))), key = lambda E: E.a_invariants()) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 5 over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 6 over Finite Field of size 7] >>> curves = curves_with_j_0(GF(Integer(25))); len(curves) 6 >>> all(not curves[i].is_isomorphic(curves[j]) for i in range(Integer(6)) for j in range(i + Integer(1), Integer(6))) True >>> set(E.j_invariant() for E in curves) {0}
For \(K=\GF{q}\) where \(q\equiv5\mod{6}\) there are two curves, quadratic twists of each other by \(-3\): \(y^2=x^3+1\) and \(y^2=x^3-27\):
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0 sage: curves_with_j_0(GF(5)) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 5] sage: curves_with_j_0(GF(11)) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 11, Elliptic Curve defined by y^2 = x^3 + 6 over Finite Field of size 11]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0 >>> curves_with_j_0(GF(Integer(5))) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 5] >>> curves_with_j_0(GF(Integer(11))) [Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 11, Elliptic Curve defined by y^2 = x^3 + 6 over Finite Field of size 11]
- sage.schemes.elliptic_curves.ell_finite_field.curves_with_j_0_char2(K)[source]¶
Return a complete list of pairwise nonisomorphic elliptic curves with \(j\)-invariant 0 over the finite field \(K\) of characteristic 2.
Note
The number of twists is either 3 or 7 depending on whether the field has odd or even degree over \(\GF{2}\). See [Connell1999], pages 429-431.
Examples:
In odd degree, there are three isomorphism classes all with representatives defined over \(\GF{2}\):
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char2 sage: # needs sage.rings.finite_rings sage: K = GF(2**7) sage: curves = curves_with_j_0_char2(K) sage: len(curves) 3 sage: [E.ainvs() for E in curves] [(0, 0, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 1, 1, 1)]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char2 >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(7)) >>> curves = curves_with_j_0_char2(K) >>> len(curves) 3 >>> [E.ainvs() for E in curves] [(0, 0, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 1, 1, 1)]
Check that the curves are mutually non-isomorphic:
sage: all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ....: for e1 in curves for e2 in curves) True
>>> from sage.all import * >>> all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ... for e1 in curves for e2 in curves) True
Check that the weight formula holds:
sage: sum(1/len(E.automorphisms()) for E in curves) == 1 # needs sage.rings.finite_rings True
>>> from sage.all import * >>> sum(Integer(1)/len(E.automorphisms()) for E in curves) == Integer(1) # needs sage.rings.finite_rings True
In even degree there are seven isomorphism classes:
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char2 sage: # needs sage.rings.finite_rings sage: K = GF(2**8) sage: curves = EllipticCurve(j=K(0)).twists() sage: len(curves) 7 sage: [E.ainvs() for E in curves] # random [(0, 0, 1, 0, 0), (0, 0, 1, 0, z8^5 + z8^4 + z8^3), (0, 0, 1, z8^6 + z8^5 + z8^2 + 1, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, z8^3 + z8^2 + 1), (0, 0, z8^6 + z8^3 + z8^2, 0, 0), (0, 0, z8^6 + z8^3 + z8^2, 0, z8^3 + z8^2)]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char2 >>> # needs sage.rings.finite_rings >>> K = GF(Integer(2)**Integer(8)) >>> curves = EllipticCurve(j=K(Integer(0))).twists() >>> len(curves) 7 >>> [E.ainvs() for E in curves] # random [(0, 0, 1, 0, 0), (0, 0, 1, 0, z8^5 + z8^4 + z8^3), (0, 0, 1, z8^6 + z8^5 + z8^2 + 1, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, 0), (0, 0, z8^4 + z8^3 + z8^2 + 1, 0, z8^3 + z8^2 + 1), (0, 0, z8^6 + z8^3 + z8^2, 0, 0), (0, 0, z8^6 + z8^3 + z8^2, 0, z8^3 + z8^2)]
Check that the twists are mutually non-isomorphic:
sage: all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ....: for e1 in curves for e2 in curves) True
>>> from sage.all import * >>> all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ... for e1 in curves for e2 in curves) True
Check that the weight formula holds:
sage: sum(1/len(E.automorphisms()) for E in curves) == 1 # needs sage.rings.finite_rings True
>>> from sage.all import * >>> sum(Integer(1)/len(E.automorphisms()) for E in curves) == Integer(1) # needs sage.rings.finite_rings True
- sage.schemes.elliptic_curves.ell_finite_field.curves_with_j_0_char3(K)[source]¶
Return a complete list of pairwise nonisomorphic elliptic curves with \(j\)-invariant 0 over the finite field \(K\) of characteristic 3.
Note
The number of twists is either 4 or 6 depending on whether the field has odd or even degree over \(\GF{3}\). See [Connell1999], pages 429-431.
Examples:
In odd degree, there are four isomorphism classes:
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char3 sage: # needs sage.rings.finite_rings sage: K = GF(3**5) sage: curves = curves_with_j_0_char3(K) sage: len(curves) 4 sage: [E.ainvs() for E in curves] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 0), (0, 0, 0, 2, z5^4 + z5^3 + z5^2), (0, 0, 0, 2, 2*z5^4 + 2*z5^3 + 2*z5^2)]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char3 >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(5)) >>> curves = curves_with_j_0_char3(K) >>> len(curves) 4 >>> [E.ainvs() for E in curves] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 0), (0, 0, 0, 2, z5^4 + z5^3 + z5^2), (0, 0, 0, 2, 2*z5^4 + 2*z5^3 + 2*z5^2)]
Check that the twists are mutually non-isomorphic:
sage: all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ....: for e1 in curves for e2 in curves) True
>>> from sage.all import * >>> all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ... for e1 in curves for e2 in curves) True
Check that the weight formula holds:
sage: sum(1/len(E.automorphisms()) for E in curves) == 1 # needs sage.rings.finite_rings True
>>> from sage.all import * >>> sum(Integer(1)/len(E.automorphisms()) for E in curves) == Integer(1) # needs sage.rings.finite_rings True
In even degree, there are six isomorphism classes:
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char3 sage: # needs sage.rings.finite_rings sage: K = GF(3**4) sage: curves = EllipticCurve(j=K(0)).twists() sage: len(curves) 6 sage: [E.ainvs() for E in curves] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, 1, 0), (0, 0, 0, 1, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, z4, 0), (0, 0, 0, z4^3, 0)]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_0_char3 >>> # needs sage.rings.finite_rings >>> K = GF(Integer(3)**Integer(4)) >>> curves = EllipticCurve(j=K(Integer(0))).twists() >>> len(curves) 6 >>> [E.ainvs() for E in curves] # random [(0, 0, 0, 1, 0), (0, 0, 0, 2, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, 1, 0), (0, 0, 0, 1, 2*z4^3 + 2*z4^2 + 2*z4 + 2), (0, 0, 0, z4, 0), (0, 0, 0, z4^3, 0)]
Check that the twists are mutually non-isomorphic:
sage: all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ....: for e1 in curves for e2 in curves) True
>>> from sage.all import * >>> all((e1 == e2 or not e1.is_isomorphic(e2)) # needs sage.rings.finite_rings ... for e1 in curves for e2 in curves) True
Check that the weight formula holds:
sage: sum(1/len(E.automorphisms()) for E in curves) == 1 # needs sage.rings.finite_rings True
>>> from sage.all import * >>> sum(Integer(1)/len(E.automorphisms()) for E in curves) == Integer(1) # needs sage.rings.finite_rings True
- sage.schemes.elliptic_curves.ell_finite_field.curves_with_j_1728(K)[source]¶
Return a complete list of pairwise nonisomorphic elliptic curves with \(j\)-invariant 1728 over the finite field \(K\).
Note
In characteristics 2 and 3 (so 0=1728) this function simply calls
curves_with_j_0_char2orcurves_with_j_0_char3. Otherwise there are either 2 or 4 curves, parametrised by \(K^*/(K^*)^4\).EXAMPLES:
For \(K=\GF{q}\) where \(q\equiv1\mod{4}\), there are four curves, the quartic twists of \(y^2=x^3+x\):
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_1728 sage: sorted(curves_with_j_1728(GF(5)), key = lambda E: E.a_invariants()) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 3*x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 4*x over Finite Field of size 5] sage: curves_with_j_1728(GF(49)) # random # needs sage.rings.finite_rings [Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + z2*x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + (z2+4)*x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + (5*z2+4)*x over Finite Field in z2 of size 7^2]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_1728 >>> sorted(curves_with_j_1728(GF(Integer(5))), key = lambda E: E.a_invariants()) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 3*x over Finite Field of size 5, Elliptic Curve defined by y^2 = x^3 + 4*x over Finite Field of size 5] >>> curves_with_j_1728(GF(Integer(49))) # random # needs sage.rings.finite_rings [Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + z2*x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + (z2+4)*x over Finite Field in z2 of size 7^2, Elliptic Curve defined by y^2 = x^3 + (5*z2+4)*x over Finite Field in z2 of size 7^2]
For \(K=\GF{q}\) where \(q\equiv3\mod{4}\), there are two curves, quadratic twists of each other by \(-1\): \(y^2=x^3+x\) and \(y^2=x^3-x\):
sage: from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_1728 sage: curves_with_j_1728(GF(7)) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 6*x over Finite Field of size 7] sage: curves_with_j_1728(GF(11)) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 11, Elliptic Curve defined by y^2 = x^3 + 10*x over Finite Field of size 11]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import curves_with_j_1728 >>> curves_with_j_1728(GF(Integer(7))) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7, Elliptic Curve defined by y^2 = x^3 + 6*x over Finite Field of size 7] >>> curves_with_j_1728(GF(Integer(11))) [Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 11, Elliptic Curve defined by y^2 = x^3 + 10*x over Finite Field of size 11]
- sage.schemes.elliptic_curves.ell_finite_field.fill_ss_j_dict()[source]¶
Fill the global cache of supersingular j-_polynomials.
This function does nothing except the first time it is called, when it fills
supersingular_j_polynomialswith precomputed values for \(p<300\). Setting the values this way avoids start-up costs.
- sage.schemes.elliptic_curves.ell_finite_field.is_j_supersingular(j, proof=True)[source]¶
Return
Trueif \(j\) is a supersingular \(j\)-invariant.INPUT:
j– finite field elementproof– boolean (default:True); ifTrue, returns a proved result. IfFalse, then a return value ofFalseis certain but a return value ofTruemay be based on a probabilistic test. See the ALGORITHM section below for more details.
OUTPUT: boolean;
Trueif \(j\) is supersingular, elseFalseALGORITHM:
For small characteristics \(p\) we check whether the \(j\)-invariant is in a precomputed list of supersingular values. Otherwise we next check the \(j\)-invariant. If \(j=0\), the curve is supersingular if and only if \(p=2\) or \(p\equiv3\pmod{4}\); if \(j=1728\), the curve is supersingular if and only if \(p=3\) or \(p\equiv2\pmod{3}\). Next, if the base field is the prime field \({\rm GF}(p)\), we check that \((p+1)P=0\) for several random points \(P\), returning
Falseif any fail: supersingular curves over \({\rm GF}(p)\) have cardinality \(p+1\). If Proof is false we now returnTrue. Otherwise we compute the cardinality and returnTrueif and only if it is divisible by \(p\).EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_finite_field import is_j_supersingular, supersingular_j_polynomials sage: [(p,[j for j in GF(p) if is_j_supersingular(j)]) for p in prime_range(30)] [(2, [0]), (3, [0]), (5, [0]), (7, [6]), (11, [0, 1]), (13, [5]), (17, [0, 8]), (19, [7, 18]), (23, [0, 3, 19]), (29, [0, 2, 25])] sage: [j for j in GF(109) if is_j_supersingular(j)] [17, 41, 43] sage: PolynomialRing(GF(109),'j')(supersingular_j_polynomials[109]).roots() [(43, 1), (41, 1), (17, 1)] sage: [p for p in prime_range(100) if is_j_supersingular(GF(p)(0))] [2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89] sage: [p for p in prime_range(100) if is_j_supersingular(GF(p)(1728))] [2, 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83] sage: [p for p in prime_range(100) if is_j_supersingular(GF(p)(123456))] [2, 3, 59, 89]
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import is_j_supersingular, supersingular_j_polynomials >>> [(p,[j for j in GF(p) if is_j_supersingular(j)]) for p in prime_range(Integer(30))] [(2, [0]), (3, [0]), (5, [0]), (7, [6]), (11, [0, 1]), (13, [5]), (17, [0, 8]), (19, [7, 18]), (23, [0, 3, 19]), (29, [0, 2, 25])] >>> [j for j in GF(Integer(109)) if is_j_supersingular(j)] [17, 41, 43] >>> PolynomialRing(GF(Integer(109)),'j')(supersingular_j_polynomials[Integer(109)]).roots() [(43, 1), (41, 1), (17, 1)] >>> [p for p in prime_range(Integer(100)) if is_j_supersingular(GF(p)(Integer(0)))] [2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89] >>> [p for p in prime_range(Integer(100)) if is_j_supersingular(GF(p)(Integer(1728)))] [2, 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83] >>> [p for p in prime_range(Integer(100)) if is_j_supersingular(GF(p)(Integer(123456)))] [2, 3, 59, 89]
- sage.schemes.elliptic_curves.ell_finite_field.special_supersingular_curve(F, q, endomorphism=None)[source]¶
Given a finite field
Fof characteristic \(p\), and optionally a positive integer \(q\) such that the Hilbert conductor of \(-q\) and \(-p\) equals \(p\), construct a “special” supersingular elliptic curve \(E\) defined overF.Such a curve
has coefficients in \(\mathbb F_p\);
has group structure \(E(\mathbb F_p) \cong \ZZ/(p+1)\) and \(E(\mathbb F_{p^2}) \cong \ZZ/(p+1) \times \ZZ/(p+1)\);
has an endomorphism \(\vartheta\) of degree \(q\) that anticommutes with the \(\mathbb F_p\)-Frobenius on \(E\).
(The significance of \(\vartheta\) is that any such endomorphism, together with the \(\mathbb F_p\)-Frobenius, generates the endomorphism algebra \(\mathrm{End}(E) \otimes \QQ\).)
The complexity grows exponentially in \(\log(q)\). Automatically chosen values of \(q\) lie in \(O((\log p)^2)\) assuming GRH.
INPUT:
F– finite field \(\mathbb F_{p^r}\)q– positive integer (optional, defaultNone)endomorphism– boolean (default:False); when set toTrue, it is required that \(2 \mid r\), and the function then additionally returns \(\vartheta\)
Warning
Due to Issue #38481, calling this function with a value of \(q\) larger than approximately \(p/4\) may currently fail. This failure will not occur for automatically chosen values of \(q\).
EXAMPLES:
sage: special_supersingular_curve(GF(1013^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2) sage: special_supersingular_curve(GF(1019^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1019^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1019^2 Via: (u,r,s,t) = (389*z2 + 241, 0, 0, 0)) sage: special_supersingular_curve(GF(1021^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2 to Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2) sage: special_supersingular_curve(GF(1031^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1031^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1031^2 Via: (u,r,s,t) = (747*z2 + 284, 0, 0, 0)) sage: special_supersingular_curve(GF(1033^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2, Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2 to Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2) sage: special_supersingular_curve(GF(1039^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1039^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1039^2 Via: (u,r,s,t) = (626*z2 + 200, 0, 0, 0)) sage: special_supersingular_curve(GF(1049^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2) sage: special_supersingular_curve(GF(1051^2), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1051^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1051^2 Via: (u,r,s,t) = (922*z2 + 129, 0, 0, 0))
>>> from sage.all import * >>> special_supersingular_curve(GF(Integer(1013)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1013^2) >>> special_supersingular_curve(GF(Integer(1019)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1019^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1019^2 Via: (u,r,s,t) = (389*z2 + 241, 0, 0, 0)) >>> special_supersingular_curve(GF(Integer(1021)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2 to Elliptic Curve defined by y^2 = x^3 + 791*x + 230 over Finite Field in z2 of size 1021^2) >>> special_supersingular_curve(GF(Integer(1031)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1031^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1031^2 Via: (u,r,s,t) = (747*z2 + 284, 0, 0, 0)) >>> special_supersingular_curve(GF(Integer(1033)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2, Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2 to Elliptic Curve defined by y^2 = x^3 + 53*x + 980 over Finite Field in z2 of size 1033^2) >>> special_supersingular_curve(GF(Integer(1039)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1039^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1039^2 Via: (u,r,s,t) = (626*z2 + 200, 0, 0, 0)) >>> special_supersingular_curve(GF(Integer(1049)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in z2 of size 1049^2) >>> special_supersingular_curve(GF(Integer(1051)**Integer(2)), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1051^2, Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 1051^2 Via: (u,r,s,t) = (922*z2 + 129, 0, 0, 0))
We can also supply a suitable value of \(q\) ourselves:
sage: special_supersingular_curve(GF(1019), q=99) Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field of size 1019 sage: special_supersingular_curve(GF(1019^2), q=99, endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2, Isogeny of degree 99 from Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2 to Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2) sage: special_supersingular_curve(GF(1013), q=99) Traceback (most recent call last): ... ValueError: invalid choice of q
>>> from sage.all import * >>> special_supersingular_curve(GF(Integer(1019)), q=Integer(99)) Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field of size 1019 >>> special_supersingular_curve(GF(Integer(1019)**Integer(2)), q=Integer(99), endomorphism=True) (Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2, Isogeny of degree 99 from Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2 to Elliptic Curve defined by y^2 = x^3 + 211*x + 808 over Finite Field in z2 of size 1019^2) >>> special_supersingular_curve(GF(Integer(1013)), q=Integer(99)) Traceback (most recent call last): ... ValueError: invalid choice of q
Note
This function makes no guarantees about the distribution of the output. The current implementation is deterministic in many cases.
ALGORITHM: [Bro2009], Algorithm 2.4
- sage.schemes.elliptic_curves.ell_finite_field.supersingular_j_polynomial(p, use_cache=True)[source]¶
Return a polynomial whose roots are the supersingular \(j\)-invariants in characteristic \(p\), other than 0, 1728.
INPUT:
p– integer; a prime numberuse_cache– boolean (default:True); use cached coefficients if they exist
ALGORITHM:
First compute H(X) whose roots are the Legendre \(\lambda\)-invariants of supersingular curves (Silverman V.4.1(b)) in characteristic \(p\). Then, using a resultant computation with the polynomial relating \(\lambda\) and \(j\) (Silverman III.1.7(b)), we recover the polynomial (in variable
j) whose roots are the \(j\)-invariants. Factors of \(j\) and \(j-1728\) are removed if present.Note
The only point of the use_cache parameter is to allow checking the precomputed coefficients.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_finite_field import supersingular_j_polynomial sage: f = supersingular_j_polynomial(67); f j^5 + 53*j^4 + 4*j^3 + 47*j^2 + 36*j + 8 sage: f.factor() (j + 1) * (j^2 + 8*j + 45) * (j^2 + 44*j + 24)
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_finite_field import supersingular_j_polynomial >>> f = supersingular_j_polynomial(Integer(67)); f j^5 + 53*j^4 + 4*j^3 + 47*j^2 + 36*j + 8 >>> f.factor() (j + 1) * (j^2 + 8*j + 45) * (j^2 + 44*j + 24)
sage: [supersingular_j_polynomial(p) for p in prime_range(30)] [1, 1, 1, 1, 1, j + 8, j + 9, j + 12, j + 4, j^2 + 2*j + 21]
>>> from sage.all import * >>> [supersingular_j_polynomial(p) for p in prime_range(Integer(30))] [1, 1, 1, 1, 1, j + 8, j + 9, j + 12, j + 4, j^2 + 2*j + 21]