# Compute invariants of quintics and sextics via ‘Ueberschiebung’¶

Todo

• Implement invariants in small positive characteristic.
• Cardona-Quer and additional invariants for classifying automorphism groups.

AUTHOR:

• Nick Alexander
sage.schemes.hyperelliptic_curves.invariants.Ueberschiebung(f, g, k)

Return the differential operator $$(f g)_k$$.

This is defined by Mestre on page 315 [Mes1991]:

$(f g)_k = \frac{(m - k)! (n - k)!}{m! n!} \left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right)^k .$

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import Ueberschiebung as ub
sage: R.<x, y> = QQ[]
sage: ub(x, y, 0)
x*y
sage: ub(x^5 + 1, x^5 + 1, 1)
0
sage: ub(x^5 + 5*x + 1, x^5 + 5*x + 1, 0)
x^10 + 10*x^6 + 2*x^5 + 25*x^2 + 10*x + 1

sage.schemes.hyperelliptic_curves.invariants.absolute_igusa_invariants_kohel(f)

Given a sextic form $$f$$, return the three absolute Igusa invariants used by Kohel [KohECHIDNA].

$$f$$ may be homogeneous in two variables or inhomogeneous in one.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import absolute_igusa_invariants_kohel
sage: R.<x> = QQ[]
sage: absolute_igusa_invariants_kohel(x^5 - 1)
(0, 0, 0)
sage: absolute_igusa_invariants_kohel(x^5 - x)
(100, -20000, -2000)


The following example can be checked against Kohel’s database [KohECHIDNA]

sage: i1, i2, i3 = absolute_igusa_invariants_kohel(-x^5 + 3*x^4 + 2*x^3 - 6*x^2 - 3*x + 1)
sage: list(map(factor, (i1, i2, i3)))
[2^2 * 3^5 * 5 * 31, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31]
sage: list(map(factor, (150660, 28343520, 9762768)))
[2^2 * 3^5 * 5 * 31, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31]

sage.schemes.hyperelliptic_curves.invariants.absolute_igusa_invariants_wamelen(f)

Given a sextic form $$f$$, return the three absolute Igusa invariants used by van Wamelen [Wam1999].

$$f$$ may be homogeneous in two variables or inhomogeneous in one.

REFERENCES:

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import absolute_igusa_invariants_wamelen
sage: R.<x> = QQ[]
sage: absolute_igusa_invariants_wamelen(x^5 - 1)
(0, 0, 0)


The following example can be checked against van Wamelen’s paper:

sage: i1, i2, i3 = absolute_igusa_invariants_wamelen(-x^5 + 3*x^4 + 2*x^3 - 6*x^2 - 3*x + 1)
sage: list(map(factor, (i1, i2, i3)))
[2^7 * 3^15, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31]

sage.schemes.hyperelliptic_curves.invariants.clebsch_invariants(f)

Given a sextic form $$f$$, return the Clebsch invariants $$(A, B, C, D)$$ of Mestre, p 317, [Mes1991].

$$f$$ may be homogeneous in two variables or inhomogeneous in one.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import clebsch_invariants
sage: R.<x, y> = QQ[]
sage: clebsch_invariants(x^6 + y^6)
(2, 2/3, -2/9, 0)
sage: R.<x> = QQ[]
sage: clebsch_invariants(x^6 + x^5 + x^4 + x^2 + 2)
(62/15, 15434/5625, -236951/140625, 229930748/791015625)

sage: magma(x^6 + 1).ClebschInvariants() # optional - magma
[ 2, 2/3, -2/9, 0 ]
sage: magma(x^6 + x^5 + x^4 + x^2 + 2).ClebschInvariants() # optional - magma
[ 62/15, 15434/5625, -236951/140625, 229930748/791015625 ]

sage.schemes.hyperelliptic_curves.invariants.clebsch_to_igusa(A, B, C, D)

Convert Clebsch invariants $$A, B, C, D$$ to Igusa invariants $$I_2, I_4, I_6, I_{10}$$.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import clebsch_to_igusa, igusa_to_clebsch
sage: clebsch_to_igusa(2, 3, 4, 5)
(-240, 17370, 231120, -103098906)
sage: igusa_to_clebsch(*clebsch_to_igusa(2, 3, 4, 5))
(2, 3, 4, 5)

sage: Cs = tuple(map(GF(31), (2, 3, 4, 5))); Cs
(2, 3, 4, 5)
sage: clebsch_to_igusa(*Cs)
(8, 10, 15, 26)
sage: igusa_to_clebsch(*clebsch_to_igusa(*Cs))
(2, 3, 4, 5)

sage.schemes.hyperelliptic_curves.invariants.differential_operator(f, g, k)

Return the differential operator $$(f g)_k$$ symbolically in the polynomial ring in dfdx, dfdy, dgdx, dgdy.

This is defined by Mestre on p 315 [Mes1991]:

$(f g)_k = \frac{(m - k)! (n - k)!}{m! n!} \left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right)^k .$

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import differential_operator
sage: R.<x, y> = QQ[]
sage: differential_operator(x, y, 0)
1
sage: differential_operator(x, y, 1)
-dfdy*dgdx + dfdx*dgdy
sage: differential_operator(x*y, x*y, 2)
1/4*dfdy^2*dgdx^2 - 1/2*dfdx*dfdy*dgdx*dgdy + 1/4*dfdx^2*dgdy^2
sage: differential_operator(x^2*y, x*y^2, 2)
1/36*dfdy^2*dgdx^2 - 1/18*dfdx*dfdy*dgdx*dgdy + 1/36*dfdx^2*dgdy^2
sage: differential_operator(x^2*y, x*y^2, 4)
1/576*dfdy^4*dgdx^4 - 1/144*dfdx*dfdy^3*dgdx^3*dgdy + 1/96*dfdx^2*dfdy^2*dgdx^2*dgdy^2 - 1/144*dfdx^3*dfdy*dgdx*dgdy^3 + 1/576*dfdx^4*dgdy^4

sage.schemes.hyperelliptic_curves.invariants.diffsymb(U, f, g)

Given a differential operator U in dfdx, dfdy, dgdx, dgdy, represented symbolically by U, apply it to f, g.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import diffsymb
sage: R.<x, y> = QQ[]
sage: S.<dfdx, dfdy, dgdx, dgdy> = QQ[]
sage: [ diffsymb(dd, x^2, y*0 + 1) for dd in S.gens() ]
[2*x, 0, 0, 0]
sage: [ diffsymb(dd, x*0 + 1, y^2) for dd in S.gens() ]
[0, 0, 0, 2*y]
sage: [ diffsymb(dd, x^2, y^2) for dd in S.gens() ]
[2*x*y^2, 0, 0, 2*x^2*y]

sage: diffsymb(dfdx + dfdy*dgdy, y*x^2, y^3)
2*x*y^4 + 3*x^2*y^2

sage.schemes.hyperelliptic_curves.invariants.diffxy(f, x, xtimes, y, ytimes)

Differentiate a polynomial f, xtimes with respect to x, and ytimes with respect to y.

EXAMPLES:

sage: R.<u, v> = QQ[]
sage: sage.schemes.hyperelliptic_curves.invariants.diffxy(u^2*v^3, u, 0, v, 0)
u^2*v^3
sage: sage.schemes.hyperelliptic_curves.invariants.diffxy(u^2*v^3, u, 2, v, 1)
6*v^2
sage: sage.schemes.hyperelliptic_curves.invariants.diffxy(u^2*v^3, u, 2, v, 2)
12*v
sage: sage.schemes.hyperelliptic_curves.invariants.diffxy(u^2*v^3 + u^4*v^4, u, 2, v, 2)
144*u^2*v^2 + 12*v

sage.schemes.hyperelliptic_curves.invariants.igusa_clebsch_invariants(f)

Given a sextic form $$f$$, return the Igusa-Clebsch invariants $$I_2, I_4, I_6, I_{10}$$ of Igusa and Clebsch [IJ1960].

$$f$$ may be homogeneous in two variables or inhomogeneous in one.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import igusa_clebsch_invariants
sage: R.<x, y> = QQ[]
sage: igusa_clebsch_invariants(x^6 + y^6)
(-240, 1620, -119880, -46656)
sage: R.<x> = QQ[]
sage: igusa_clebsch_invariants(x^6 + x^5 + x^4 + x^2 + 2)
(-496, 6220, -955932, -1111784)

sage: magma(x^6 + 1).IgusaClebschInvariants() # optional - magma
[ -240, 1620, -119880, -46656 ]
sage: magma(x^6 + x^5 + x^4 + x^2 + 2).IgusaClebschInvariants() # optional - magma
[ -496, 6220, -955932, -1111784 ]

sage.schemes.hyperelliptic_curves.invariants.igusa_to_clebsch(I2, I4, I6, I10)

Convert Igusa invariants $$I_2, I_4, I_6, I_{10}$$ to Clebsch invariants $$A, B, C, D$$.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import clebsch_to_igusa, igusa_to_clebsch
sage: igusa_to_clebsch(-2400, 173700, 23112000, -10309890600)
(20, 342/5, 2512/5, 43381012/1125)
sage: clebsch_to_igusa(*igusa_to_clebsch(-2400, 173700, 23112000, -10309890600))
(-2400, 173700, 23112000, -10309890600)

sage: Is = tuple(map(GF(31), (-2400, 173700, 23112000, -10309890600))); Is
(18, 7, 12, 27)
sage: igusa_to_clebsch(*Is)
(20, 25, 25, 12)
sage: clebsch_to_igusa(*igusa_to_clebsch(*Is))
(18, 7, 12, 27)

sage.schemes.hyperelliptic_curves.invariants.ubs(f)

Given a sextic form $$f$$, return a dictionary of the invariants of Mestre, p 317 [Mes1991].

$$f$$ may be homogeneous in two variables or inhomogeneous in one.

EXAMPLES:

sage: from sage.schemes.hyperelliptic_curves.invariants import ubs
sage: x = QQ['x'].0
sage: ubs(x^6 + 1)
{'A': 2,
'B': 2/3,
'C': -2/9,
'D': 0,
'Delta': -2/3*x^2*h^2,
'f': x^6 + h^6,
'i': 2*x^2*h^2,
'y1': 0,
'y2': 0,
'y3': 0}

sage: R.<u, v> = QQ[]
sage: ubs(u^6 + v^6)
{'A': 2,
'B': 2/3,
'C': -2/9,
'D': 0,
'Delta': -2/3*u^2*v^2,
'f': u^6 + v^6,
'i': 2*u^2*v^2,
'y1': 0,
'y2': 0,
'y3': 0}

sage: R.<t> = GF(31)[]
sage: ubs(t^6 + 2*t^5 + t^2 + 3*t + 1)
{'A': 0,
'B': -12,
'C': -15,
'D': -15,
'Delta': -10*t^4 + 12*t^3*h + 7*t^2*h^2 - 5*t*h^3 + 2*h^4,
'f': t^6 + 2*t^5*h + t^2*h^4 + 3*t*h^5 + h^6,
'i': -4*t^4 + 10*t^3*h + 2*t^2*h^2 - 9*t*h^3 - 7*h^4,
'y1': 4*t^2 - 10*t*h - 13*h^2,
'y2': 6*t^2 - 4*t*h + 2*h^2,
'y3': 4*t^2 - 4*t*h - 9*h^2}
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