# HG changeset patch # User Nick Alexander # Date 1241728646 25200 # Node ID b7d5265cc1f15e081318e49b494073a3813cc4cd # Parent 268d1efbf60f6c24ad9c49cb9c91ff0437e42340 [mq]: trac_6004-odd-degree-model.patch diff -r 268d1efbf60f -r b7d5265cc1f1 sage/schemes/hyperelliptic_curves/all.py --- a/sage/schemes/hyperelliptic_curves/all.py Tue May 05 23:19:43 2009 -0700 +++ b/sage/schemes/hyperelliptic_curves/all.py Thu May 07 13:37:26 2009 -0700 @@ -1,4 +1,7 @@ from constructor import HyperellipticCurve from hyperelliptic_generic import is_HyperellipticCurve from kummer_surface import KummerSurface - +from invariants import (igusa_clebsch_invariants, + absolute_igusa_invariants_kohel, + absolute_igusa_invariants_wamelen, + clebsch_invariants) diff -r 268d1efbf60f -r b7d5265cc1f1 sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py --- a/sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py Tue May 05 23:19:43 2009 -0700 +++ b/sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py Thu May 07 13:37:26 2009 -0700 @@ -7,12 +7,11 @@ import hyperelliptic_generic import jacobian_g2 +import invariants + from sage.schemes.generic.projective_space import ProjectiveSpace class HyperellipticCurve_g2_generic(hyperelliptic_generic.HyperellipticCurve_generic): - def igusa_invariants(self): - raise NotImplementedError - def is_odd_degree(self): """ Returns True if the curve is an odd degree model. @@ -26,7 +25,7 @@ else: a0 = f.leading_coefficient() c0 = h.leading_coefficient() - return (c0^2 + 4*a0) == 0 + return (c0**2 + 4*a0) == 0 def jacobian(self): return jacobian_g2.HyperellipticJacobian_g2(self) @@ -47,3 +46,219 @@ J = self.jacobian() K = J.kummer_surface() return self._kummer_morphism + + def clebsch_invariants(self): + r""" + Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: f = x^5 - x^4 + 3 + sage: HyperellipticCurve(f).clebsch_invariants() + (0, -2048/375, -4096/25, -4881645568/84375) + sage: HyperellipticCurve(f(2*x)).clebsch_invariants() + (0, -8388608/375, -1073741824/25, -5241627016305836032/84375) + + sage: HyperellipticCurve(f, x).clebsch_invariants() + (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625) + sage: HyperellipticCurve(f(2*x), 2*x).clebsch_invariants() + (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625) + + TESTS:: + + sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma + [ 0, -2048/375, -4096/25, -4881645568/84375 ] + sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma + [ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ] + sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma + [ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ] + sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma + [ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ] + """ + f, h = self.hyperelliptic_polynomials() + return invariants.clebsch_invariants(4*f + h**2) + + def igusa_clebsch_invariants(self): + r""" + Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: f = x^5 - x + 2 + sage: HyperellipticCurve(f).igusa_clebsch_invariants() + (-640, -20480, 1310720, 52160364544) + sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants() + (-40960, -83886080, 343597383680, 56006764965979488256) + + sage: HyperellipticCurve(f, x).igusa_clebsch_invariants() + (-640, 17920, -1966656, 52409511936) + sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants() + (-40960, 73400320, -515547070464, 56274284941110411264) + + TESTS:: + + sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma + [ -640, -20480, 1310720, 52160364544 ] + sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma + [ -40960, -83886080, 343597383680, 56006764965979488256 ] + sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma + [ -640, 17920, -1966656, 52409511936 ] + sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma + [ -40960, 73400320, -515547070464, 56274284941110411264 ] + """ + f, h = self.hyperelliptic_polynomials() + return invariants.igusa_clebsch_invariants(4*f + h**2) + + def absolute_igusa_invariants_wamelen(self): + r""" + Return the three absolute Igusa invariants used by van Wamelen [W]_. + + EXAMPLES:: + + sage: R. = QQ[] + sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_wamelen() + (0, 0, 0) + sage: HyperellipticCurve((x^5 - 1)(x - 2), (x^2)(x - 2)).absolute_igusa_invariants_wamelen() + (0, 0, 0) + """ + f, h = self.hyperelliptic_polynomials() + return invariants.absolute_igusa_invariants_wamelen(4*f + h**2) + + def absolute_igusa_invariants_kohel(self): + r""" + Return the three absolute Igusa invariants used by Kohel [K]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_kohel() + (0, 0, 0) + sage: HyperellipticCurve(x^5 - x + 1, x^2).absolute_igusa_invariants_kohel() + (-1030567/178769, 259686400/178769, 20806400/178769) + sage: HyperellipticCurve((x^5 - x + 1)(3*x + 1), (x^2)(3*x + 1)).absolute_igusa_invariants_kohel() + (-1030567/178769, 259686400/178769, 20806400/178769) + """ + f, h = self.hyperelliptic_polynomials() + return invariants.absolute_igusa_invariants_kohel(4*f + h**2) + + def clebsch_invariants(self): + r""" + Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: f = x^5 - x^4 + 3 + sage: HyperellipticCurve(f).clebsch_invariants() + (0, -2048/375, -4096/25, -4881645568/84375) + sage: HyperellipticCurve(f(2*x)).clebsch_invariants() + (0, -8388608/375, -1073741824/25, -5241627016305836032/84375) + + sage: HyperellipticCurve(f, x).clebsch_invariants() + (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625) + sage: HyperellipticCurve(f(2*x), 2*x).clebsch_invariants() + (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625) + + TESTS:: + + sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma + [ 0, -2048/375, -4096/25, -4881645568/84375 ] + sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma + [ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ] + sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma + [ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ] + sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma + [ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ] + """ + f, h = self.hyperelliptic_polynomials() + return invariants.clebsch_invariants(4*f + h**2) + + def igusa_clebsch_invariants(self): + r""" + Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: f = x^5 - x + 2 + sage: HyperellipticCurve(f).igusa_clebsch_invariants() + (-640, -20480, 1310720, 52160364544) + sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants() + (-40960, -83886080, 343597383680, 56006764965979488256) + + sage: HyperellipticCurve(f, x).igusa_clebsch_invariants() + (-640, 17920, -1966656, 52409511936) + sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants() + (-40960, 73400320, -515547070464, 56274284941110411264) + + TESTS:: + + sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma + [ 0, -2048/375, -4096/25, -4881645568/84375 ] + sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma + [ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ] + sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma + [ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ] + sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma + [ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ] + """ + f, h = self.hyperelliptic_polynomials() + return invariants.igusa_clebsch_invariants(4*f + h**2) + + def absolute_igusa_invariants_wamelen(self): + r""" + Return the three absolute Igusa invariants used by van Wamelen [W]_. + + EXAMPLES:: + + sage: R. = QQ[] + sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_wamelen() + (0, 0, 0) + sage: HyperellipticCurve((x^5 - 1)(x - 2), (x^2)(x - 2)).absolute_igusa_invariants_wamelen() + (0, 0, 0) + """ + f, h = self.hyperelliptic_polynomials() + return invariants.absolute_igusa_invariants_wamelen(4*f + h**2) + + def absolute_igusa_invariants_kohel(self): + r""" + Return the three absolute Igusa invariants used by Kohel [K]_. + + SEEALSO:: + + .. sage.schemes.hyperelliptic_curves.invariants + + EXAMPLES:: + + sage: R. = QQ[] + sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_kohel() + (0, 0, 0) + sage: HyperellipticCurve(x^5 - x + 1, x^2).absolute_igusa_invariants_kohel() + (-1030567/178769, 259686400/178769, 20806400/178769) + sage: HyperellipticCurve((x^5 - x + 1)(3*x + 1), (x^2)(3*x + 1)).absolute_igusa_invariants_kohel() + (-1030567/178769, 259686400/178769, 20806400/178769) + """ + f, h = self.hyperelliptic_polynomials() + return invariants.absolute_igusa_invariants_kohel(4*f + h**2) diff -r 268d1efbf60f -r b7d5265cc1f1 sage/schemes/hyperelliptic_curves/hyperelliptic_generic.py --- a/sage/schemes/hyperelliptic_curves/hyperelliptic_generic.py Tue May 05 23:19:43 2009 -0700 +++ b/sage/schemes/hyperelliptic_curves/hyperelliptic_generic.py Thu May 07 13:37:26 2009 -0700 @@ -63,8 +63,10 @@ names = ["x","y"] elif isinstance(names,str): names = names.split(",") + self._names = names P1 = PolynomialRing(R,name=names[0]) P2 = PolynomialRing(P1,name=names[1]) + self._PP = PP self._printing_ring = P2 self._hyperelliptic_polynomials = (f,h) self._genus = genus diff -r 268d1efbf60f -r b7d5265cc1f1 sage/schemes/hyperelliptic_curves/invariants.py --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/sage/schemes/hyperelliptic_curves/invariants.py Thu May 07 13:37:26 2009 -0700 @@ -0,0 +1,366 @@ +r""" +Compute invariants of quintics and sextics via ``Ueberschiebung''. + +REFERENCES:: + + .. [M] Mestre, Jean-Francois. Construction de courbes de genre $2$ a + partir de leurs modules. (French) [Constructing genus-$2$ curves from + their moduli] Effective methods in algebraic geometry (Castiglioncello, + 1990), 313--334, Progr. Math., 94, Birkhauser Boston, Boston, MA, 1991. + + .. [I] Igusa, Jun-ichi. Arithmetic variety of moduli for genus two. + Ann. of Math. (2) 72 1960 612--649. + +TODO:: + + * Implement invariants in small positive characteristic. + + * Cardona-Quer and additional invariants for classifying automorphism groups. + +AUTHOR:: + + * Nick Alexander +""" + +from sage.rings.all import ZZ, QQ +from sage.rings.all import PolynomialRing + +def diffxy(f, x, xtimes, y, ytimes): + r""" + Differentiate a polynomial ``f``, ``xtimes`` with respect to ``x``, and + ```ytimes`` with respect to ``y``. + + EXAMPLES:: + sage: R. = 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 + """ + h = f + for i in range(xtimes): h = h.derivative(x) + for j in range(ytimes): h = h.derivative(y) + return h + +def differential_operator(f, g, k): + r""" + 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 [M]_: + `(f g)_k = \frac{(m - k)! (n - k)!}{m! n!} \left( + \frac{\del f}{\del x} \frac{\del g}{\del y} - + \frac{\del f}{\del y} \frac{\del g}{\del x} \right)^k ` . + + EXAMPLES:: + + sage: from sage.schemes.hyperelliptic_curves.invariants import differential_operator + sage: R. = 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 + """ + (x, y) = f.parent().gens() + n = max(ZZ(f.degree()), ZZ(k)) + m = max(ZZ(g.degree()), ZZ(k)) + R, (fx, fy, gx, gy) = PolynomialRing(f.base_ring(), 4, 'dfdx,dfdy,dgdx,dgdy').objgens() + const = (m - k).factorial() * (n - k).factorial() / (m.factorial() * n.factorial()) + U = f.base_ring()(const) * (fx*gy - fy*gx)**k + return U + +def diffsymb(U, f, g): + r""" + 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. = QQ[] + sage: S. = 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 + """ + (x, y) = f.parent().gens() + R, (fx, fy, gx, gy) = PolynomialRing(f.base_ring(), 4, 'dfdx,dfdy,dgdx,dgdy').objgens() + res = 0 + for coeff, mon in list(U): + mon = R(mon) + a = diffxy(f, x, mon.degree(fx), y, mon.degree(fy)) + b = diffxy(g, x, mon.degree(gx), y, mon.degree(gy)) + temp = coeff * a * b + res = res + temp + return res + +def Ueberschiebung(f, g, k): + r""" + Return the differential operator `(f g)_k`. + + This is defined by Mestre on page 315: + `(f g)_k = \frac{(m - k)! (n - k)!}{m! n!} \left( + \frac{\del f}{\del x} \frac{\del g}{\del y} - + \frac{\del f}{\del y} \frac{\del g}{\del x} \right)^k ` . + + EXAMPLES:: + + sage: from sage.schemes.hyperelliptic_curves.invariants import Ueberschiebung as ub + sage: R. = 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 + """ + U = differential_operator(f, g, k) + # U is the (f g)_k = ... of Mestre, p315, symbolically + return diffsymb(U, f, g) + +def ubs(f): + r""" + Given a sextic form `f`, return a dictionary of the invariants of Mestre, p 317 [M]_. + + `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, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': x^6 + h^6, 'i': 2*x^2*h^2, 'Delta': -2/3*x^2*h^2, 'y1': 0, 'y3': 0, 'y2': 0} + + sage: R. = QQ[] + sage: ubs(u^6 + v^6) + {'A': 2, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': u^6 + v^6, 'i': 2*u^2*v^2, 'Delta': -2/3*u^2*v^2, 'y1': 0, 'y3': 0, 'y2': 0} + + sage: R. = GF(31)[] + sage: ubs(t^6 + 2*t^5 + t^2 + 3*t + 1) + {'A': 0, 'C': -15, 'B': -12, 'D': -15, '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, 'Delta': -10*t^4 + 12*t^3*h + 7*t^2*h^2 - 5*t*h^3 + 2*h^4, 'y1': 4*t^2 - 10*t*h - 13*h^2, 'y3': 4*t^2 - 4*t*h - 9*h^2, 'y2': 6*t^2 - 4*t*h + 2*h^2} + """ + ub = Ueberschiebung + if f.parent().ngens() == 1: + f = PolynomialRing(f.parent().base_ring(), 1, f.parent().variable_name())(f) + x1, x2 = f.homogenize().parent().gens() + f = sum([ f[i]*x1**i*x2**(6-i) for i in range(7) ]) + U = {} + U['f'] = f + U['i'] = ub(f, f, 4) + U['Delta'] = ub(U['i'], U['i'], 2) + U['y1'] = ub(f, U['i'], 4) + U['y2'] = ub(U['i'], U['y1'], 2) + U['y3'] = ub(U['i'], U['y2'], 2) + U['A'] = ub(f, f, 6) + U['B'] = ub(U['i'], U['i'], 4) + U['C'] = ub(U['i'], U['Delta'], 4) + U['D'] = ub(U['y3'], U['y1'], 2) + return U + +def clebsch_to_igusa(A, B, C, D): + r""" + 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) + """ + I2 = -120*A + I4 = -720*A**2 + 6750*B + I6 = 8640*A**3 - 108000*A*B + 202500*C + I10 = -62208*A**5 + 972000*A**3*B + 1620000*A**2*C - 3037500*A*B**2 - 6075000*B*C - 4556250*D + return (I2, I4, I6, I10) + +def igusa_to_clebsch(I2, I4, I6, I10): + r""" + 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) + """ + A = -(+ I2) / 120 + B = -(- I2**2 - 20*I4)/135000 + C = -(+ I2**3 + 80*I2*I4 - 600*I6)/121500000 + D = -(+ 9*I2**5 + 700*I2**3*I4 - 3600*I2**2*I6 - 12400*I2*I4**2 + 48000*I4*I6 + 10800000*I10) / 49207500000000 + return (A, B, C, D) + + +def clebsch_invariants(f): + r""" + Given a sextic form `f`, return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_. + + `f` may be homogeneous in two variables or inhomogeneous in one. + + EXAMPLES:: + + sage: R. = QQ[] + sage: clebsch_invariants(x^6 + y^6) + (2, 2/3, -2/9, 0) + sage: R. = 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 ] + """ + R = f.parent().base_ring() + if R.characteristic() in [2, 3, 5]: + raise NotImplementedError, "Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5" + + U = ubs(f) + L = U['A'], U['B'], U['C'], U['D'] + assert all(t.is_constant() for t in L) + return tuple([ t.constant_coefficient() for t in L ]) + +def igusa_clebsch_invariants(f): + r""" + Given a sextic form `f`, return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_. + + `f` may be homogeneous in two variables or inhomogeneous in one. + + EXAMPLES:: + + sage: R. = QQ[] + sage: igusa_clebsch_invariants(x^6 + y^6) + (-240, 1620, -119880, -46656) + sage: R. = 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 ] + + TESTS:: + + Let's check a symbolic example: + + sage: R. = QQ[] + sage: S. = R[] + sage: igusa_clebsch_invariants(x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e)[0] + 6*b^2 - 16*a*c + 40*d + + sage: absolute_igusa_invariants_wamelen(GF(5)['x'](x^6 - 2*x)) + Traceback (most recent call last): + ... + NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5 + """ + return clebsch_to_igusa(*clebsch_invariants(f)) + +def absolute_igusa_invariants_wamelen(f): + r""" + Given a sextic form `f`, return the three absolute Igusa invariants used by van Wamelen [W]_. + + `f` may be homogeneous in two variables or inhomogeneous in one. + + REFERENCES:: + + .. [W] van Wamelen, Paul. Examples of genus two CM curves defined + over the rationals. + Math. Comp. 68 (1999), no. 225, 307--320. + + EXAMPLES:: + + sage: R. = 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: map(factor, (i1, i2, i3)) + [2^7 * 3^15, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31] + + TESTS:: + + sage: absolute_igusa_invariants_wamelen(GF(3)['x'](x^5 - 2*x)) + Traceback (most recent call last): + ... + NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5 + """ + I2, I4, I6, I10 = igusa_clebsch_invariants(f) + i1 = I2**5/I10 + i2 = I2**3*I4/I10 + i3 = I2**2*I6/I10 + return (i1, i2, i3) + +def absolute_igusa_invariants_kohel(f): + r""" + Given a sextic form `f`, return the three absolute Igusa invariants used by Kohel [K]_. + + `f` may be homogeneous in two variables or inhomogeneous in one. + + REFERENCES:: + + .. [K] Kohel, David. ECHIDNA: Databases for Elliptic Curves and Higher Dimensional Analogues. + Available at http://echidna.maths.usyd.edu.au/~kohel/dbs/ + + EXAMPLES:: + + sage: R. = 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 [K]_: + + sage: i1, i2, i3 = absolute_igusa_invariants_kohel(-x^5 + 3*x^4 + 2*x^3 - 6*x^2 - 3*x + 1) + sage: map(factor, (i1, i2, i3)) + [2^2 * 3^5 * 5 * 31, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31] + sage: map(factor, (150660, 28343520, 9762768)) + [2^2 * 3^5 * 5 * 31, 2^5 * 3^11 * 5, 2^4 * 3^9 * 31] + + TESTS:: + + sage: absolute_igusa_invariants_kohel(GF(2)['x'](x^5 - x)) + Traceback (most recent call last): + ... + NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5 + """ + I2, I4, I6, I10 = igusa_clebsch_invariants(f) + i1 = I4*I6/I10 + i2 = I2**3*I4/I10 + i3 = I2**2*I6/I10 + return (i1, i2, i3)