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: h = -x^5 + 3*x^4 + 2*x^3 - 6*x^2 - 3*x + 1 sage: i1, i2, i3 = absolute_igusa_invariants_kohel(h) 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: h = -x^5 + 3*x^4 + 2*x^3 - 6*x^2 - 3*x + 1 sage: i1, i2, i3 = absolute_igusa_invariants_wamelen(h) 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
Uindfdx, dfdy, dgdx, dgdy, represented symbolically byU, apply it tof, 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,xtimeswith respect tox, and`ytimeswith respect toy.EXAMPLES:
sage: from sage.schemes.hyperelliptic_curves.invariants import diffxy sage: R.<u, v> = QQ[] sage: diffxy(u^2*v^3, u, 0, v, 0) u^2*v^3 sage: diffxy(u^2*v^3, u, 2, v, 1) 6*v^2 sage: diffxy(u^2*v^3, u, 2, v, 2) 12*v sage: 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}