def velu(kernel, domain=None):
E = kernel[0].curve()
Q = PolynomialRing(E.base_field(), ['x', 'y'])
x, y = Q.gens()
X = x
Y = y
R = []
v = 0
w = 0
for P in kernel:
g_x = 3 * P[0]**2 + E.a4()
g_y = -2 * P[1]
u_P = g_y**2
if 2 * P == E(0, 1, 0):
R.append(P)
v_P = g_x
else:
if not -P in R:
R.append(P)
v_P = 2 * g_x
else:
continue
v += v_P
w += (u_P + P[0] * v_P)
X += (v_P / (x - P[0]) + u_P / (x - P[0])**2)
Y -= (2 * u_P * y / (x - P[0])**3 + v_P * (y - P[1]) / (x - P[0])**2 -
g_x * g_y / (x - P[0])**2)
a = E.a4() - 5 * v
b = E.a6() - 7 * w
if domain != None:
try:
a = domain.base_field()(a)
b = domain.base_field()(b)
codomain = EllipticCurve(domain.base_field(), [a, b])
R = PolynomialRing(E.base_field(), ['x', 'y'])
Rf = R.fraction_field()
X = Rf(X)
Y = Rf(Y)
f = standard_form((X, Y), domain)
except:
codomain = EllipticCurve(E.base_field(), [a, b])
f = standard_form((X, Y), E)
else:
codomain = EllipticCurve(E.base_field(), [a, b])
f = standard_form((X, Y), E)
return codomain, f
def mod5family(a, b):
"""
Formulas for computing the family of elliptic curves with
congruent mod-5 representation.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.mod5family import mod5family
sage: mod5family(0,1)
Elliptic Curve defined by y^2 = x^3 + (t^30+30*t^29+435*t^28+4060*t^27+27405*t^26+142506*t^25+593775*t^24+2035800*t^23+5852925*t^22+14307150*t^21+30045015*t^20+54627300*t^19+86493225*t^18+119759850*t^17+145422675*t^16+155117520*t^15+145422675*t^14+119759850*t^13+86493225*t^12+54627300*t^11+30045015*t^10+14307150*t^9+5852925*t^8+2035800*t^7+593775*t^6+142506*t^5+27405*t^4+4060*t^3+435*t^2+30*t+1) over Fraction Field of Univariate Polynomial Ring in t over Rational Field
"""
J = 4 * a**3 / (4 * a**3 + 27 * b**2)
alpha = [0 for _ in range(21)]
alpha[0] = 1
alpha[1] = 0
alpha[2] = 190 * (J - 1)
alpha[3] = -2280 * (J - 1)**2
alpha[4] = 855 * (J - 1)**2 * (-17 + 16 * J)
alpha[5] = 3648 * (J - 1)**3 * (17 - 9 * J)
alpha[6] = 11400 * (J - 1)**3 * (17 - 8 * J)
alpha[7] = -27360 * (J - 1)**4 * (17 + 26 * J)
alpha[8] = 7410 * (J - 1)**4 * (-119 - 448 * J + 432 * J**2)
alpha[9] = 79040 * (J - 1)**5 * (17 + 145 * J - 108 * J**2)
alpha[10] = 8892 * (J - 1)**5 * (187 + 2640 * J - 5104 * J**2 +
1152 * J**3)
alpha[11] = 98800 * (J - 1)**6 * (-17 - 388 * J + 864 * J**2)
alpha[12] = 7410 * (J - 1)**6 * (-187 - 6160 * J + 24464 * J**2 -
24192 * J**3)
alpha[13] = 54720 * (J - 1)**7 * (17 + 795 * J - 3944 * J**2 + 9072 * J**3)
alpha[14] = 2280 * (J - 1)**7 * (221 + 13832 * J - 103792 * J**2 +
554112 * J**3 - 373248 * J**4)
alpha[15] = 1824 * (J - 1)**8 * (-119 - 9842 * J + 92608 * J**2 -
911520 * J**3 + 373248 * J**4)
alpha[16] = 4275 * (J - 1)**8 * (-17 - 1792 * J + 23264 * J**2 -
378368 * J**3 + 338688 * J**4)
alpha[17] = 18240 * (J - 1)**9 * (1 + 133 * J - 2132 * J**2 +
54000 * J**3 - 15552 * J**4)
alpha[18] = 190 * (J - 1)**9 * (17 + 2784 * J - 58080 * J**2 + 2116864 *
J**3 - 946944 * J**4 + 2985984 * J**5)
alpha[19] = 360 * (J - 1)**10 * (-1 + 28 * J - 1152 * J**2) * (
1 + 228 * J + 176 * J**2 + 1728 * J**3)
alpha[20] = (J - 1)**10 * (-19 - 4560 * J + 144096 * J**2 -
9859328 * J**3 - 8798976 * J**4 -
226934784 * J**5 + 429981696 * J**6)
beta = [0 for _ in range(31)]
beta[0] = 1
beta[1] = 30
beta[2] = -435 * (J - 1)
beta[3] = 580 * (J - 1) * (-7 + 9 * J)
beta[4] = 3915 * (J - 1)**2 * (7 - 8 * J)
beta[5] = 1566 * (J - 1)**2 * (91 - 78 * J + 48 * J**2)
beta[6] = -84825 * (J - 1)**3 * (7 + 16 * J)
beta[7] = 156600 * (J - 1)**3 * (-13 - 91 * J + 92 * J**2)
beta[8] = 450225 * (J - 1)**4 * (13 + 208 * J - 144 * J**2)
beta[9] = 100050 * (J - 1)**4 * (143 + 4004 * J - 5632 * J**2 +
1728 * J**3)
beta[10] = 30015 * (J - 1)**5 * (-1001 - 45760 * J + 44880 * J**2 -
6912 * J**3)
beta[11] = 600300 * (J - 1)**5 * (-91 - 6175 * J + 9272 * J**2 -
2736 * J**3)
beta[12] = 950475 * (J - 1)**6 * (91 + 8840 * J - 7824 * J**2)
beta[13] = 17108550 * (J - 1)**6 * (7 + 926 * J - 1072 * J**2 + 544 * J**3)
beta[14] = 145422675 * (J - 1)**7 * (-1 - 176 * J + 48 * J**2 - 384 * J**3)
beta[15] = 155117520 * (J - 1)**8 * (1 + 228 * J + 176 * J**2 +
1728 * J**3)
beta[16] = 145422675 * (J - 1)**8 * (1 + 288 * J + 288 * J**2 +
5120 * J**3 - 6912 * J**4)
beta[17] = 17108550 * (J - 1)**8 * (7 + 2504 * J + 3584 * J**2 + 93184 *
J**3 - 283392 * J**4 + 165888 * J**5)
beta[18] = 950475 * (J - 1)**9 * (-91 - 39936 * J - 122976 * J**2 -
2960384 * J**3 + 11577600 * J**4 -
5971968 * J**5)
beta[19] = 600300 * (J - 1)**9 * (-91 - 48243 * J - 191568 * J**2 -
6310304 * J**3 + 40515072 * J**4 -
46455552 * J**5 + 11943936 * J**6)
beta[20] = 30015 * (J - 1)**10 * (1001 + 634920 * J + 3880800 * J**2 +
142879744 * J**3 - 1168475904 * J**4 +
1188919296 * J**5 - 143327232 * J**6)
beta[21] = 100050 * (J - 1)**10 * (143 + 107250 * J + 808368 * J**2 +
38518336 * J**3 - 451953408 * J**4 +
757651968 * J**5 - 367276032 * J**6)
beta[22] = 450225 * (J - 1)**11 * (-13 - 11440 * J - 117216 * J**2 -
6444800 * J**3 + 94192384 * J**4 -
142000128 * J**5 + 95551488 * J**6)
beta[23] = 156600 * (J - 1)**11 * (
-13 - 13299 * J - 163284 * J**2 - 11171552 * J**3 + 217203840 * J**4 -
474406656 * J**5 + 747740160 * J**6 - 429981696 * J**7)
beta[24] = 6525*(J - 1)**12*(91 + 107536*J + 1680624*J**2 + 132912128*J**3 -\
3147511552*J**4 + 6260502528*J**5 - 21054173184*J**6 + 10319560704*J**7)
beta[25] = 1566*(J - 1)**12*(91 + 123292*J + 2261248*J**2 + 216211904*J**3 - \
6487793920*J**4 + 17369596928*J**5 - 97854234624*J**6 + 96136740864*J**7 - 20639121408*J**8)
beta[26] = 3915*(J - 1)**13*(-7 - 10816*J - 242352*J**2 - 26620160*J**3 + 953885440*J**4 - \
2350596096*J**5 + 26796552192*J**6 - 13329432576*J**7)
beta[27] = 580*(J - 1)**13*(-7 - 12259*J - 317176*J**2 - 41205008*J**3 + \
1808220160*J**4 - 5714806016*J**5 + 93590857728*J**6 - 70131806208*J**7 - 36118462464*J**8)
beta[28] = 435*(J - 1)**14*(1 + 1976*J + 60720*J**2 + 8987648*J**3 - 463120640*J**4 + 1359157248*J**5 - \
40644882432*J**6 - 5016453120*J**7 + 61917364224*J**8)
beta[29] = 30*(J - 1)**14*(1 + 2218*J + 77680*J**2 + 13365152*J**3 - \
822366976*J**4 + 2990693888*J**5 - 118286217216*J**6 - 24514928640*J**7 + 509958291456*J**8 - 743008370688*J**9)
beta[30] = (J - 1)**15*(-1 - 2480*J - 101040*J**2 - 19642496*J**3 + 1399023872*J**4 - \
4759216128*J**5 + 315623485440*J**6 + 471904911360*J**7 - 2600529297408*J**8 + 8916100448256*J**9)
R = PolynomialRing(QQ, 't')
c2 = a * R(alpha)
c3 = b * R(beta)
d = c2.denominator().lcm(c3.denominator())
F = R.fraction_field()
return EllipticCurve(F, [c2 * d**4, c3 * d**6])
