Descent on elliptic curves over \(\QQ\) with a 2-isogeny#
- sage.schemes.elliptic_curves.descent_two_isogeny.test_els(a, b, c, d, e)[source]#
Doctest function for cdef int everywhere_locally_soluble(mpz_t, mpz_t, mpz_t, mpz_t, mpz_t).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import test_els sage: for _ in range(1000): ....: a,b,c,d,e = randint(1,1000), randint(1,1000), randint(1,1000), randint(1,1000), randint(1,1000) ....: if pari.Pol([a,b,c,d,e]).hyperellratpoints(1000, 1): ....: try: ....: if not test_els(a,b,c,d,e): ....: print("This never happened", a, b, c, d, e) ....: except ValueError: ....: continue
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import test_els >>> for _ in range(Integer(1000)): ... a,b,c,d,e = randint(Integer(1),Integer(1000)), randint(Integer(1),Integer(1000)), randint(Integer(1),Integer(1000)), randint(Integer(1),Integer(1000)), randint(Integer(1),Integer(1000)) ... if pari.Pol([a,b,c,d,e]).hyperellratpoints(Integer(1000), Integer(1)): ... try: ... if not test_els(a,b,c,d,e): ... print("This never happened", a, b, c, d, e) ... except ValueError: ... continue
- sage.schemes.elliptic_curves.descent_two_isogeny.test_padic_square(a, p)[source]#
Doctest function for cdef int padic_square(mpz_t, unsigned long).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import test_padic_square as ps sage: for i in [1..300]: ....: for p in prime_range(100): ....: if Qp(p)(i).is_square() != bool(ps(i,p)): ....: print(i, p)
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import test_padic_square as ps >>> for i in (ellipsis_range(Integer(1),Ellipsis,Integer(300))): ... for p in prime_range(Integer(100)): ... if Qp(p)(i).is_square() != bool(ps(i,p)): ... print(i, p)
- sage.schemes.elliptic_curves.descent_two_isogeny.test_qpls(a, b, c, d, e, p)[source]#
Testing function for Qp_soluble.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import test_qpls as tq sage: tq(1,2,3,4,5,7) 1
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import test_qpls as tq >>> tq(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(7)) 1
- sage.schemes.elliptic_curves.descent_two_isogeny.test_valuation(a, p)[source]#
Doctest function for cdef long valuation(mpz_t, mpz_t).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import test_valuation as tv sage: for i in [1..20]: ....: print('{:>10} {} {} {}'.format(str(factor(i)), tv(i,2), tv(i,3), tv(i,5))) 1 0 0 0 2 1 0 0 3 0 1 0 2^2 2 0 0 5 0 0 1 2 * 3 1 1 0 7 0 0 0 2^3 3 0 0 3^2 0 2 0 2 * 5 1 0 1 11 0 0 0 2^2 * 3 2 1 0 13 0 0 0 2 * 7 1 0 0 3 * 5 0 1 1 2^4 4 0 0 17 0 0 0 2 * 3^2 1 2 0 19 0 0 0 2^2 * 5 2 0 1
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import test_valuation as tv >>> for i in (ellipsis_range(Integer(1),Ellipsis,Integer(20))): ... print('{:>10} {} {} {}'.format(str(factor(i)), tv(i,Integer(2)), tv(i,Integer(3)), tv(i,Integer(5)))) 1 0 0 0 2 1 0 0 3 0 1 0 2^2 2 0 0 5 0 0 1 2 * 3 1 1 0 7 0 0 0 2^3 3 0 0 3^2 0 2 0 2 * 5 1 0 1 11 0 0 0 2^2 * 3 2 1 0 13 0 0 0 2 * 7 1 0 0 3 * 5 0 1 1 2^4 4 0 0 17 0 0 0 2 * 3^2 1 2 0 19 0 0 0 2^2 * 5 2 0 1
- sage.schemes.elliptic_curves.descent_two_isogeny.two_descent_by_two_isogeny(E, global_limit_small=10, global_limit_large=10000, verbosity=0, selmer_only=0, proof=1)[source]#
Given an elliptic curve E with a two-isogeny phi : E –> E’ and dual isogeny phi’, runs a two-isogeny descent on E, returning n1, n2, n1’ and n2’. Here n1 is the number of quartic covers found with a rational point, and n2 is the number which are ELS.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny sage: E = EllipticCurve('14a') sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 0 sage: E = EllipticCurve('65a') sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 1 sage: # needs sage.symbolic sage: x,y = var('x,y') sage: E = EllipticCurve(y^2 == x^3 + x^2 - 25*x + 39) sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 2 sage: E = EllipticCurve(y^2 + x*y + y == x^3 - 131*x + 558) sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 3
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny >>> E = EllipticCurve('14a') >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 0 >>> E = EllipticCurve('65a') >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 1 >>> # needs sage.symbolic >>> x,y = var('x,y') >>> E = EllipticCurve(y**Integer(2) == x**Integer(3) + x**Integer(2) - Integer(25)*x + Integer(39)) >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 2 >>> E = EllipticCurve(y**Integer(2) + x*y + y == x**Integer(3) - Integer(131)*x + Integer(558)) >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 3
Using the verbosity option:
sage: E = EllipticCurve('14a') sage: two_descent_by_two_isogeny(E, verbosity=1) 2-isogeny Results: 2 <= #E(Q)/phi'(E'(Q)) <= 2 2 <= #E'(Q)/phi(E(Q)) <= 2 #Sel^(phi')(E'/Q) = 2 #Sel^(phi)(E/Q) = 2 1 <= #Sha(E'/Q)[phi'] <= 1 1 <= #Sha(E/Q)[phi] <= 1 1 <= #Sha(E/Q)[2], #Sha(E'/Q)[2] <= 1 0 <= rank of E(Q) = rank of E'(Q) <= 0 (2, 2, 2, 2)
>>> from sage.all import * >>> E = EllipticCurve('14a') >>> two_descent_by_two_isogeny(E, verbosity=Integer(1)) 2-isogeny Results: 2 <= #E(Q)/phi'(E'(Q)) <= 2 2 <= #E'(Q)/phi(E(Q)) <= 2 #Sel^(phi')(E'/Q) = 2 #Sel^(phi)(E/Q) = 2 1 <= #Sha(E'/Q)[phi'] <= 1 1 <= #Sha(E/Q)[phi] <= 1 1 <= #Sha(E/Q)[2], #Sha(E'/Q)[2] <= 1 0 <= rank of E(Q) = rank of E'(Q) <= 0 (2, 2, 2, 2)
Handling curves whose discriminants involve larger than wordsize primes:
sage: E = EllipticCurve('14a') sage: E = E.quadratic_twist(next_prime(10^20)) sage: E Elliptic Curve defined by y^2 = x^3 + x^2 + 716666666666666667225666666666666666775672*x - 391925925925925926384240370370370370549019837037037037060249356 over Rational Field sage: E.discriminant().factor() -1 * 2^18 * 7^3 * 100000000000000000039^6 sage: log(100000000000000000039.0, 2.0) 66.438... sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 0
>>> from sage.all import * >>> E = EllipticCurve('14a') >>> E = E.quadratic_twist(next_prime(Integer(10)**Integer(20))) >>> E Elliptic Curve defined by y^2 = x^3 + x^2 + 716666666666666667225666666666666666775672*x - 391925925925925926384240370370370370549019837037037037060249356 over Rational Field >>> E.discriminant().factor() -1 * 2^18 * 7^3 * 100000000000000000039^6 >>> log(RealNumber('100000000000000000039.0'), RealNumber('2.0')) 66.438... >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 0
- sage.schemes.elliptic_curves.descent_two_isogeny.two_descent_by_two_isogeny_work(c, d, global_limit_small=10, global_limit_large=10000, verbosity=0, selmer_only=0, proof=1)[source]#
Do all the work in doing a two-isogeny descent.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny_work sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(13,128) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 0 sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(1,-16) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 1 sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(10,8) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 2 sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(85,320) sage: log(n1,2) + log(n1_prime,2) - 2 # the rank 3
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny_work >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(Integer(13),Integer(128)) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 0 >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(Integer(1),-Integer(16)) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 1 >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(Integer(10),Integer(8)) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 2 >>> n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny_work(Integer(85),Integer(320)) >>> log(n1,Integer(2)) + log(n1_prime,Integer(2)) - Integer(2) # the rank 3