CoCalc Public Files18.783 Lecture 6 Division polynomials.ipynbOpen with one click!
Author: Andrew Sutherland
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Description: Code to symbolically compute division polynomials, both directly from the group law, and via recurrences.
Compute Environment: Ubuntu 20.04 (Default)
In [1]:
R.<x,y,A,B>=PolynomialRing(Rationals()) def reduce_poly_once(f): g = R(0) for t in f.monomials(): if t.mod(y^2) == 0: g += f.monomial_coefficient(t)*R(t/y^2*(x^3+A*x+B)) else: g += f.monomial_coefficient(t)*t return g def reduce_poly(f): while f.degrees()[1] > 1: f = reduce_poly_once(f) return f def reduce_point(P): return (reduce_poly(P[0]),reduce_poly(P[1]),reduce_poly(P[2])) def dbl(P): x1= P[0]; y1=P[1]; z1=P[2] x3=x1^4-2*A*x1^2*z1^4-8*B*x1*z1^6+A^2*z1^8 y3=x1^6+5*A*x1^4*z1^4+20*B*x1^3*z1^6-5*A^2*x1^2*z1^8-4*A*B*x1*z1^10-(A^3+8*B^2)*z1^12 z3=2*y1*z1 return reduce_point((x3,y3,z3)) def add(P,Q): if compare_point(P,Q): return dbl(P) x1=P[0]; y1=P[1]; z1=P[2]; x2=Q[0]; y2=Q[1]; z2=Q[2] x3 = A*(x1*z2^2+x2*z1^2)*z1^2*z2^2 + 2*B*z1^4*z2^4 + x1^2*x2*z2^2+x1*x2^2*z1^2-2*y1*y2*z1*z2 y3 =(A*x2*z2+B*z2^3)*y2*z1^6 - (A*x1*z1+B*z1^3)*y1*z2^6+2*(x2^3*y1*z1^3 - x1^3*y2*z2^3) + 3*x1*x2*z1*z2*(x1*y2*z1 - x2*y1*z2) + 3*y1*y2*(y1*z2^3 - y2*z1^3) z3 = x1*z2^2-x2*z1^2 return reduce_point((x3,y3,z3)) def compare_point(P,Q): return (P[0]*Q[2]^2 == Q[0]*P[2]^2 and P[1]*Q[2]^3 == Q[1]*P[2]^3)
In [2]:
# compute the first 7 division polynomials using the group law # NB: to avoid introducing extra factors compute P[2m+1] as P[m]+P[m+1] and P[2m]=P[m]+P[m] # It will also work with arbitrary additions, but this may require removing common factors from the coordinates # A more complicated version of reduce_point that does this is provided below. P = [(R(0),R(1),R(0))] # point at infinity P.append((R(x),R(y),R(1))) # generic point P P.append(dbl(P[1])) # 2P = P + P P.append(add(P[2],P[1])) # 3P = 2P + P P.append(dbl(P[2])) # 4P = 2P + 2P P.append(add(P[2],P[3])) # 5P = 2P + 3P P.append(dbl(P[3])) # 6P = 3P + 3P P.append(add(P[3],P[4])) # 7P = 3P + 4P
In [4]:
for n in range(1,8): print("%d, %s, %s, %s"%(n,P[n][0], P[n][1], P[n][2]))
1, x, y, 1 2, x^4 - 2*x^2*A + A^2 - 8*x*B, x^6 + 5*x^4*A - 5*x^2*A^2 + 20*x^3*B - A^3 - 4*x*A*B - 8*B^2, 2*y 3, x^9 - 12*x^7*A + 30*x^5*A^2 - 96*x^6*B + 36*x^3*A^3 - 24*x^4*A*B + 9*x*A^4 + 48*x^2*A^2*B + 48*x^3*B^2 + 8*A^3*B + 96*x*A*B^2 + 64*B^3, -x^12*y - 22*x^10*y*A + 165*x^8*y*A^2 - 220*x^9*y*B + 92*x^6*y*A^3 + 528*x^7*y*A*B + 185*x^4*y*A^4 - 264*x^5*y*A^2*B + 1776*x^6*y*B^2 + 90*x^2*y*A^5 + 80*x^3*y*A^3*B + 960*x^4*y*A*B^2 + 3*y*A^6 + 132*x*y*A^4*B + 624*x^2*y*A^2*B^2 + 320*x^3*y*B^3 + 96*y*A^3*B^2 + 896*x*y*A*B^3 + 512*y*B^4, -3*x^4 - 6*x^2*A + A^2 - 12*x*B 4, x^16 - 40*x^14*A + 348*x^12*A^2 - 544*x^13*B + 1000*x^10*A^3 + 128*x^11*A*B + 1478*x^8*A^4 + 1760*x^9*A^2*B + 2944*x^10*B^2 + 1000*x^6*A^5 + 3584*x^7*A^3*B + 9696*x^8*A*B^2 + 348*x^4*A^6 + 1952*x^5*A^4*B + 14208*x^6*A^2*B^2 + 8704*x^7*B^3 - 40*x^2*A^7 + 1408*x^3*A^5*B + 2368*x^4*A^3*B^2 + 26624*x^5*A*B^3 + A^8 - 96*x*A^6*B + 1536*x^2*A^4*B^2 + 7680*x^3*A^2*B^3 + 15872*x^4*B^4 - 32*A^5*B^2 + 11264*x^2*A*B^4 - 256*A^2*B^4 + 4096*x*B^5, x^24 + 68*x^22*A - 1694*x^20*A^2 + 1232*x^21*B - 3276*x^18*A^3 - 9856*x^19*A*B - 19601*x^16*A^4 + 10032*x^17*A^2*B - 58688*x^18*B^2 - 63352*x^14*A^5 + 26368*x^15*A^3*B - 150960*x^16*A*B^2 - 86436*x^12*A^6 - 140000*x^13*A^4*B - 278400*x^14*A^2*B^2 - 49408*x^15*B^3 - 63352*x^10*A^7 - 290304*x^11*A^5*B - 804160*x^12*A^3*B^2 - 492800*x^13*A*B^3 - 19601*x^8*A^8 - 293536*x^9*A^6*B - 894720*x^10*A^4*B^2 - 2248960*x^11*A^2*B^3 - 439040*x^12*B^4 - 3276*x^6*A^9 - 87808*x^7*A^7*B - 848928*x^8*A^5*B^2 - 2213120*x^9*A^3*B^3 - 3591680*x^10*A*B^4 - 1694*x^4*A^10 + 4368*x^5*A^8*B - 318080*x^6*A^6*B^2 - 1391360*x^7*A^4*B^3 - 4753920*x^8*A^2*B^4 - 2111488*x^9*B^5 + 68*x^2*A^11 - 7040*x^3*A^9*B + 20160*x^4*A^7*B^2 - 517888*x^5*A^5*B^3 - 2293760*x^6*A^3*B^4 - 5320704*x^7*A*B^5 + A^12 + 112*x*A^10*B - 8640*x^2*A^8*B^2 - 29440*x^3*A^6*B^3 - 224000*x^4*A^4*B^4 - 3698688*x^5*A^2*B^5 - 1949696*x^6*B^6 + 80*A^9*B^2 - 3840*x*A^7*B^3 - 84480*x^2*A^5*B^4 - 4096*x^3*A^3*B^5 - 2953216*x^4*A*B^6 - 55296*x*A^4*B^5 - 155648*x^2*A^2*B^6 - 851968*x^3*B^7 - 8192*A^3*B^6 - 163840*x*A*B^7 - 32768*B^8, 4*x^6*y + 20*x^4*y*A - 20*x^2*y*A^2 + 80*x^3*y*B - 4*y*A^3 - 16*x*y*A*B - 32*y*B^2 5, x^25 - 100*x^23*A + 2250*x^21*A^2 - 2080*x^22*B + 11900*x^19*A^3 + 4664*x^20*A*B + 45415*x^17*A^4 + 28880*x^18*A^2*B + 50320*x^19*B^2 + 85624*x^15*A^5 + 163000*x^16*A^3*B + 301920*x^17*A*B^2 + 110060*x^13*A^6 + 240640*x^14*A^4*B + 1172928*x^15*A^2*B^2 + 367040*x^16*B^3 + 47000*x^11*A^7 + 558960*x^12*A^5*B + 888320*x^13*A^3*B^2 + 3278080*x^14*A*B^3 - 27425*x^9*A^8 + 504672*x^10*A^6*B + 931680*x^11*A^4*B^2 + 4373760*x^12*A^2*B^3 + 2924800*x^13*B^4 - 16020*x^7*A^9 - 60880*x^8*A^7*B + 1420480*x^9*A^5*B^2 + 2282240*x^10*A^3*B^3 + 8104960*x^11*A*B^4 + 2794*x^5*A^10 - 130400*x^6*A^8*B + 157120*x^7*A^6*B^2 + 1436800*x^8*A^4*B^3 + 7705600*x^9*A^2*B^4 + 3491840*x^10*B^5 + 620*x^3*A^11 + 1880*x^4*A^9*B - 275712*x^5*A^7*B^2 - 247040*x^6*A^5*B^3 + 2944000*x^7*A^3*B^4 + 7987200*x^8*A*B^5 + 25*x*A^12 + 1360*x^2*A^10*B + 4240*x^3*A^8*B^2 - 362240*x^4*A^6*B^3 - 1483520*x^5*A^4*B^4 + 4915200*x^6*A^2*B^5 + 2805760*x^7*B^6 + 24*A^11*B + 1760*x*A^9*B^2 + 11520*x^2*A^7*B^3 - 307200*x^3*A^5*B^4 - 2406400*x^4*A^3*B^5 + 3842048*x^5*A*B^6 + 960*A^8*B^3 + 17920*x*A^6*B^4 - 153600*x^2*A^4*B^5 - 2088960*x^3*A^2*B^6 + 1064960*x^4*B^7 + 10240*A^5*B^5 - 1146880*x^2*A*B^7 + 32768*A^2*B^7 - 327680*x*B^8, x^36*y + 162*x^34*y*A - 10659*x^32*y*A^2 + 4692*x^33*y*B - 17680*x^30*y*A^3 - 107712*x^31*y*A*B - 636140*x^28*y*A^4 + 556512*x^29*y*A^2*B - 884544*x^30*y*B^2 - 4387464*x^26*y*A^5 + 677056*x^27*y*A^3*B - 2780928*x^28*y*A*B^2 - 13183308*x^24*y*A^6 - 17228880*x^25*y*A^4*B - 20537280*x^26*y*A^2*B^2 + 1880320*x^27*y*B^3 - 25646064*x^22*y*A^7 - 64421568*x^23*y*A^5*B - 150297984*x^24*y*A^3*B^2 - 51922944*x^25*y*A*B^3 - 25306290*x^20*y*A^8 - 184798176*x^21*y*A^6*B - 321867072*x^22*y*A^4*B^2 - 621909504*x^23*y*A^2*B^3 - 94222080*x^24*y*B^4 - 18498420*x^18*y*A^9 - 182074176*x^19*y*A^7*B - 932327424*x^20*y*A^5*B^2 - 1202617344*x^21*y*A^3*B^3 - 1621002240*x^22*y*A*B^4 - 21703866*x^16*y*A^10 - 40481928*x^17*y*A^8*B - 1191254976*x^18*y*A^6*B^2 - 2559863040*x^19*y*A^4*B^3 - 4509453312*x^20*y*A^2*B^4 - 1437769728*x^21*y*B^5 - 21670512*x^14*y*A^11 - 27662400*x^15*y*A^9*B - 536307840*x^16*y*A^7*B^2 - 3179197440*x^17*y*A^5*B^3 - 8099312640*x^18*y*A^3*B^4 - 7632285696*x^19*y*A*B^5 - 6610076*x^12*y*A^12 - 140843232*x^13*y*A^10*B + 101454912*x^14*y*A^8*B^2 - 2536363008*x^15*y*A^6*B^3 - 5723815680*x^16*y*A^4*B^4 - 20244584448*x^17*y*A^2*B^5 - 3534606336*x^18*y*B^6 + 2411000*x^10*y*A^13 - 91460544*x^11*y*A^11*B - 247137024*x^12*y*A^9*B^2 - 808531968*x^13*y*A^7*B^3 - 2914099200*x^14*y*A^5*B^4 - 16995778560*x^15*y*A^3*B^5 - 21804982272*x^16*y*A*B^6 + 983460*x^8*y*A^14 + 3901616*x^9*y*A^12*B - 336004416*x^10*y*A^10*B^2 - 618893568*x^11*y*A^8*B^3 - 2617466880*x^12*y*A^6*B^4 - 2851430400*x^13*y*A^4*B^5 - 30747820032*x^14*y*A^2*B^6 - 8883929088*x^15*y*B^7 + 43632*x^6*y*A^15 + 5757888*x^7*y*A^13*B - 4558464*x^8*y*A^11*B^2 - 724111360*x^9*y*A^9*B^3 - 2336937984*x^10*y*A^7*B^4 - 2465390592*x^11*y*A^5*B^5 - 9487220736*x^12*y*A^3*B^6 - 25508708352*x^13*y*A*B^7 + 13369*x^4*y*A^16 - 41760*x^5*y*A^14*B + 18446400*x^6*y*A^12*B^2 - 8517120*x^7*y*A^10*B^3 - 1189704960*x^8*y*A^8*B^4 - 5621059584*x^9*y*A^6*B^5 - 1191321600*x^10*y*A^4*B^6 - 16594698240*x^11*y*A^2*B^7 - 6868500480*x^12*y*B^8 + 1122*x^2*y*A^17 + 23104*x^3*y*A^15*B + 127488*x^4*y*A^13*B^2 + 28750848*x^5*y*A^11*B^3 + 101713920*x^6*y*A^9*B^4 - 1808105472*x^7*y*A^7*B^5 - 8285405184*x^8*y*A^5*B^6 - 4022992896*x^9*y*A^3*B^7 - 9923198976*x^10*y*A*B^8 + 5*y*A^18 + 2004*x*y*A^16*B + 49344*x^2*y*A^14*B^2 + 305408*x^3*y*A^12*B^3 + 29445120*x^4*y*A^10*B^4 + 273530880*x^5*y*A^8*B^5 - 1746468864*x^6*y*A^6*B^6 - 10423369728*x^7*y*A^4*B^7 - 5163319296*x^8*y*A^2*B^8 - 1853358080*x^9*y*B^9 + 1152*y*A^15*B^2 + 68608*x*y*A^13*B^3 + 568320*x^2*y*A^11*B^4 + 19845120*x^3*y*A^9*B^5 + 342589440*x^4*y*A^7*B^6 - 665321472*x^5*y*A^5*B^7 - 10914103296*x^6*y*A^3*B^8 - 2548039680*x^7*y*A*B^9 + 35584*y*A^12*B^4 + 829440*x*y*A^10*B^5 + 9400320*x^2*y*A^8*B^6 + 246743040*x^3*y*A^6*B^7 + 448266240*x^4*y*A^4*B^8 - 8024752128*x^5*y*A^2*B^9 - 497025024*x^6*y*B^10 + 450560*y*A^9*B^6 + 5898240*x*y*A^7*B^7 + 100270080*x^2*y*A^5*B^8 + 697303040*x^3*y*A^3*B^9 - 3605004288*x^4*y*A*B^10 + 2949120*y*A^6*B^8 + 30146560*x*y*A^4*B^9 + 336592896*x^2*y*A^2*B^10 - 742391808*x^3*y*B^11 + 10485760*y*A^3*B^10 + 75497472*x*y*A*B^11 + 16777216*y*B^12, 5*x^12 + 62*x^10*A - 105*x^8*A^2 + 380*x^9*B - 300*x^6*A^3 + 240*x^7*A*B - 125*x^4*A^4 - 696*x^5*A^2*B - 240*x^6*B^2 - 50*x^2*A^5 - 80*x^3*A^3*B - 1920*x^4*A*B^2 + A^6 - 100*x*A^4*B - 240*x^2*A^2*B^2 - 1600*x^3*B^3 - 32*A^3*B^2 - 640*x*A*B^3 - 256*B^4 6, x^36 - 210*x^34*A + 10137*x^32*A^2 - 6216*x^33*B + 88272*x^30*A^3 + 42240*x^31*A*B + 714996*x^28*A^4 + 269760*x^29*A^2*B + 475392*x^30*B^2 + 2606472*x^26*A^5 + 3688448*x^27*A^3*B + 4805952*x^28*A*B^2 + 7456644*x^24*A^6 + 8489376*x^25*A^4*B + 39484800*x^26*A^2*B^2 + 8163328*x^27*B^3 + 7190448*x^22*A^7 + 55653120*x^23*A^5*B + 41125824*x^24*A^3*B^2 + 160694784*x^25*A*B^3 - 8116242*x^20*A^8 + 115594560*x^21*A^6*B + 133544448*x^22*A^4*B^2 + 364216320*x^23*A^2*B^3 + 209995776*x^24*B^4 - 19901036*x^18*A^9 + 14737920*x^19*A^7*B + 447236928*x^20*A^5*B^2 + 533376000*x^21*A^3*B^3 + 992489472*x^22*A*B^4 - 8116242*x^16*A^10 - 181155888*x^17*A^8*B + 338391168*x^18*A^6*B^2 + 741335040*x^19*A^4*B^3 + 2530340352*x^20*A^2*B^4 + 452689920*x^21*B^5 + 7190448*x^14*A^11 - 172455168*x^15*A^9*B - 402744384*x^16*A^7*B^2 + 43587072*x^17*A^5*B^3 + 3223127040*x^18*A^3*B^4 + 3155927040*x^19*A*B^5 + 7456644*x^12*A^12 - 22640064*x^13*A^10*B - 615036672*x^14*A^8*B^2 - 1557872640*x^15*A^6*B^3 - 373547520*x^16*A^4*B^4 + 6567948288*x^17*A^2*B^5 + 1560477696*x^18*B^6 + 2606472*x^10*A^13 + 30200832*x^11*A^11*B - 138437184*x^12*A^9*B^2 - 1671936000*x^13*A^7*B^3 - 4962889728*x^14*A^5*B^4 + 95256576*x^15*A^3*B^5 + 6510845952*x^16*A*B^6 + 714996*x^8*A^14 + 9733280*x^9*A^12*B + 112791168*x^10*A^10*B^2 - 458158080*x^11*A^8*B^3 - 3559228416*x^12*A^6*B^4 - 11318280192*x^13*A^4*B^5 + 1350696960*x^14*A^2*B^6 + 2157576192*x^15*B^7 + 88272*x^6*A^15 + 3341568*x^7*A^13*B + 27139392*x^8*A^11*B^2 + 285068800*x^9*A^9*B^3 - 759146496*x^10*A^7*B^4 - 6944145408*x^11*A^5*B^5 - 16939892736*x^12*A^3*B^6 + 20054016*x^13*A*B^7 + 10137*x^4*A^16 + 247488*x^5*A^14*B + 10742784*x^6*A^12*B^2 + 53035008*x^7*A^10*B^3 + 549688320*x^8*A^8*B^4 - 627892224*x^9*A^6*B^5 - 11101372416*x^10*A^4*B^6 - 18262130688*x^11*A^2*B^7 - 795672576*x^12*B^8 - 210*x^2*A^17 + 41472*x^3*A^15*B + 426432*x^4*A^13*B^2 + 21089280*x^5*A^11*B^3 + 110714880*x^6*A^9*B^4 + 679182336*x^7*A^7*B^5 + 577732608*x^8*A^5*B^6 - 14597750784*x^9*A^3*B^7 - 12533760000*x^10*A*B^8 + A^18 - 456*x*A^16*B + 54144*x^2*A^14*B^2 + 983040*x^3*A^12*B^3 + 23827968*x^4*A^10*B^4 + 217202688*x^5*A^8*B^5 + 552665088*x^6*A^6*B^6 + 2051407872*x^7*A^4*B^7 - 14164623360*x^8*A^2*B^8 - 3636985856*x^9*B^9 - 192*A^15*B^2 + 23040*x*A^13*B^3 + 1360896*x^2*A^11*B^4 + 20619264*x^3*A^9*B^5 + 258293760*x^4*A^7*B^6 + 624033792*x^5*A^5*B^7 + 1568931840*x^6*A^3*B^8 - 7656701952*x^7*A*B^9 + 1536*A^12*B^4 + 749568*x*A^10*B^5 + 17006592*x^2*A^8*B^6 + 189923328*x^3*A^6*B^7 + 772669440*x^4*A^4*B^8 + 286261248*x^5*A^2*B^9 - 1572864000*x^6*B^10 + 122880*A^9*B^6 + 8650752*x*A^7*B^7 + 110886912*x^2*A^5*B^8 + 566231040*x^3*A^3*B^9 + 31457280*x^4*A*B^10 + 1572864*A^6*B^8 + 47185920*x*A^4*B^9 + 276824064*x^2*A^2*B^10 + 50331648*x^3*B^11 + 8388608*A^3*B^10 + 100663296*x*A*B^11 + 16777216*B^12, x^54 + 333*x^52*A - 46665*x^50*A^2 + 14004*x^51*B - 61965*x^48*A^3 - 700740*x^49*A*B - 11598930*x^46*A^4 + 8653680*x^47*A^2*B - 7962120*x^48*B^2 - 154158282*x^44*A^5 + 7026960*x^45*A^3*B - 35128800*x^46*A*B^2 - 922228038*x^42*A^6 - 917319672*x^43*A^4*B - 771391296*x^44*A^2*B^2 + 86023680*x^45*B^3 - 3845062014*x^40*A^7 - 6314686056*x^41*A^5*B - 11379356640*x^42*A^3*B^2 - 2776542336*x^43*A*B^3 - 8570651061*x^38*A^8 - 39903599376*x^39*A^6*B - 45895195440*x^40*A^4*B^2 - 67017944448*x^41*A^2*B^3 - 7753091328*x^42*B^4 - 15210522769*x^36*A^9 - 89511234288*x^37*A^7*B - 300298487328*x^38*A^5*B^2 - 242621487360*x^39*A^3*B^3 - 255793135104*x^40*A*B^4 - 40009201755*x^34*A^10 - 60875531028*x^35*A^8*B - 892213931136*x^36*A^6*B^2 - 1187584600320*x^37*A^4*B^3 - 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802816*x^3*B^7 + 24576*A^3*B^6 + 229376*x*A*B^7 + 65536*B^8
In [5]:
# compute first 7 division polynomials using recurrence relations M=8 psi=[R(0) for m in range(0,M+2)] for m in range(0,5): psi[m]=P[m][2] psi[3]=-psi[3] # fix sign of psi_3 to match definition # note that we need psi[m+2] to compute omega[m] for n in range(5,M+2): m = n//2 if n==2*m: psi[2*m] =reduce_poly(R(psi[m]*(psi[m+2]*psi[m-1]^2-psi[m-2]*psi[m+1]^2)/(2*y))) else: psi[2*m+1]=reduce_poly(psi[m+2]*psi[m]^3-psi[m-1]*psi[m+1]^3) phi=[R(0) for m in range(0,M)] for m in range(1,M): phi[m] = reduce_poly(x*psi[m]^2-psi[m+1]*psi[m-1]) omega=[R(0) for m in range(0,M)] omega[0]=1 omega[1]=y for m in range(2,M): omega[m] = reduce_poly(R((psi[m+2]*psi[m-1]^2-psi[m-2]*psi[m+1]^2)/(4*y)))
In [8]:
# verify that we get the same result using the recurrences as when we used the group law for m in range(1,M): print("%d: %s"%(m,compare_point(P[m],(phi[m],omega[m],psi[m]))))
1: True 2: True 3: True 4: True 5: True 6: True 7: True
In [9]:
P[7][2]
7*x^24 + 308*x^22*A - 2954*x^20*A^2 + 3944*x^21*B - 19852*x^18*A^3 - 112*x^19*A*B - 35231*x^16*A^4 - 92568*x^17*A^2*B - 42896*x^18*B^2 - 82264*x^14*A^5 - 31808*x^15*A^3*B - 571872*x^16*A*B^2 - 111916*x^12*A^6 - 161840*x^13*A^4*B - 615360*x^14*A^2*B^2 - 829696*x^15*B^3 - 42168*x^10*A^7 - 608160*x^11*A^5*B - 297472*x^12*A^3*B^2 - 2132480*x^13*A*B^3 + 15673*x^8*A^8 - 425712*x^9*A^6*B - 1192800*x^10*A^4*B^2 - 2603776*x^11*A^2*B^3 - 928256*x^12*B^4 + 14756*x^6*A^9 - 53824*x^7*A^7*B - 831936*x^8*A^5*B^2 - 3727360*x^9*A^3*B^3 - 3293696*x^10*A*B^4 + 1302*x^4*A^10 + 57288*x^5*A^8*B - 190400*x^6*A^6*B^2 - 1314560*x^7*A^4*B^3 - 7069440*x^8*A^2*B^4 - 1555456*x^9*B^5 + 196*x^2*A^11 + 1680*x^3*A^9*B + 134400*x^4*A^7*B^2 - 168448*x^5*A^5*B^3 - 2293760*x^6*A^3*B^4 - 7127040*x^7*A*B^5 - A^12 + 392*x*A^10*B + 3696*x^2*A^8*B^2 + 152320*x^3*A^6*B^3 + 394240*x^4*A^4*B^4 - 3698688*x^5*A^2*B^5 - 2809856*x^6*B^6 + 160*A^9*B^2 + 7168*x*A^7*B^3 + 96768*x^2*A^5*B^4 + 831488*x^3*A^3*B^5 - 3039232*x^4*A*B^6 + 3328*A^6*B^4 + 64512*x*A^4*B^5 + 544768*x^2*A^2*B^6 - 802816*x^3*B^7 + 24576*A^3*B^6 + 229376*x*A*B^7 + 65536*B^8
In [10]:
# this is a more thorough point reduction function that removes common factors def reduce_point(P): x3 = reduce_poly(P[0]) y3 = reduce_poly(P[1]) z3 = reduce_poly(P[2]) z32 = reduce_poly(z3^2) z33 = reduce_poly(z3^3) h = R((y3*numerator(x3/z32)) / (x3*numerator(y3/z33))) x3 = R(x3/h^2); y3 = R(y3/h^3); z3 = R(z3/h) if z3.mod(y) == 0 and x3.mod(x^3+A*x+B) == 0 and y3.mod((x^3+A*x+B)^2) == 0: z3 = R(z3/y); x3 = R(x3/(x^3+A*x+B)); y3 = R(y*y3/(x^3+A*x+B)^2) if z3.mod(x^3+A*x+B) == 0 and x3.mod(x^3+A*x+B) == 0 and y3.mod((x^3+A*x+B)*y) == 0: z3 = R(y*z3/(x^3+A*x+B)); x3 = R(x3/(x^3+A*x+B)); y3 = R(y3/(y*(x^3+A*x+B))) xc = gcd([int(c) for c in x3.coefficients()]) yc = gcd([int(c) for c in y3.coefficients()]) zc = gcd([int(c) for c in z3.coefficients()]) if zc%2 == 0 and xc%4 == 0 and yc%8 == 0: z3 = R(z3/2); x3 = R(x3/4); y3 = R(y3/8); xc = gcd([int(c) for c in x3.coefficients()]) yc = gcd([int(c) for c in y3.coefficients()]) zc = gcd([int(c) for c in z3.coefficients()]) if zc%2 == 0 and xc%4 == 0 and yc%8 == 0: z3 = R(z3/2); x3 = R(x3/4); y3 = R(y3/8); return (x3,y3,z3)
In [12]:
print(P[7][1])
x^72*y + 612*x^70*y*A - 162978*x^68*y*A^2 + 35304*x^69*y*B + 292468*x^66*y*A^3 - 3434544*x^67*y*A*B - 136715887*x^64*y*A^4 + 89908104*x^65*y*A^2*B - 50917584*x^66*y*B^2 - 2878540896*x^62*y*A^5 - 200586496*x^63*y*A^3*B - 189519264*x^64*y*A*B^2 - 27433746480*x^60*y*A^6 - 26018484672*x^61*y*A^4*B - 16463183616*x^62*y*A^2*B^2 + 1762855424*x^63*y*B^3 - 199034507872*x^58*y*A^7 - 246004159104*x^59*y*A^5*B - 406415978496*x^60*y*A^3*B^2 - 85929177600*x^61*y*A*B^3 - 609871362252*x^56*y*A^8 - 3109591425472*x^57*y*A^6*B - 2039624365440*x^58*y*A^4*B^2 - 3320266882560*x^59*y*A^2*B^3 - 324496236288*x^60*y*B^4 - 2174973922000*x^54*y*A^9 - 7471676126976*x^55*y*A^7*B - 33552575179008*x^56*y*A^5*B^2 - 12713923320320*x^57*y*A^3*B^3 - 17693603873280*x^58*y*A*B^4 - 14130630520248*x^52*y*A^10 + 4101348572640*x^53*y*A^8*B - 130922115667200*x^54*y*A^6*B^2 - 169642743723520*x^55*y*A^4*B^3 - 125014623694080*x^56*y*A^2*B^4 - 30593851180032*x^57*y*B^5 - 68900083189392*x^50*y*A^11 - 8605433289024*x^51*y*A^9*B - 217926712194048*x^52*y*A^7*B^2 - 632259170357760*x^53*y*A^5*B^3 - 1064741840757760*x^54*y*A^3*B^4 - 436341351665664*x^55*y*A*B^5 - 167073184622428*x^48*y*A^12 - 708249198446496*x^49*y*A^10*B + 251792366015808*x^50*y*A^8*B^2 - 2313505685601792*x^51*y*A^6*B^3 - 2023621737603840*x^52*y*A^4*B^4 - 5310527812356096*x^53*y*A^2*B^5 - 300860200374272*x^54*y*B^6 - 91360535934752*x^46*y*A^13 - 3432590319777024*x^47*y*A^11*B - 1510401726798720*x^48*y*A^9*B^2 - 5617556690680320*x^49*y*A^7*B^3 - 2839612341671424*x^50*y*A^5*B^4 - 15966902375829504*x^51*y*A^3*B^5 - 10499159812128768*x^52*y*A*B^6 + 523260349630896*x^44*y*A^14 - 6319832437196096*x^45*y*A^12*B - 19246254403993344*x^46*y*A^10*B^2 - 19775991018769920*x^47*y*A^8*B^3 - 20022971241857280*x^48*y*A^6*B^4 - 14067513340627968*x^49*y*A^4*B^5 - 53537876183273472*x^50*y*A^2*B^6 - 8755093109587968*x^51*y*B^7 + 1553783847153120*x^42*y*A^15 - 1462692262238592*x^43*y*A^13*B - 52561037092497408*x^44*y*A^11*B^2 - 105937656830343680*x^45*y*A^9*B^3 - 134205196037990400*x^46*y*A^7*B^4 - 30925995255521280*x^47*y*A^5*B^5 - 102913273858596864*x^48*y*A^3*B^6 - 77563938707767296*x^49*y*A*B^7 + 2202358583484334*x^40*y*A^16 + 14305027505731776*x^41*y*A^14*B - 39106894162292352*x^42*y*A^12*B^2 - 289086484679849472*x^43*y*A^10*B^3 - 577759130084770560*x^44*y*A^8*B^4 - 511772340052611072*x^45*y*A^6*B^5 - 129845846052691968*x^46*y*A^4*B^6 - 282127071306645504*x^47*y*A^2*B^7 - 21064201313255424*x^48*y*B^8 + 2075223517438872*x^38*y*A^17 + 27568828118888704*x^39*y*A^15*B + 83848236937690368*x^40*y*A^13*B^2 - 288150785984709120*x^41*y*A^11*B^3 - 1307489312312578560*x^42*y*A^9*B^4 - 2587819974973415424*x^43*y*A^7*B^5 - 1412854420406796288*x^44*y*A^5*B^6 - 739059060674002944*x^45*y*A^3*B^7 - 175647413834219520*x^46*y*A*B^8 + 1816604592687668*x^36*y*A^18 + 24188364321167856*x^37*y*A^16*B + 233740320180806400*x^38*y*A^14*B^2 + 290912493768020480*x^39*y*A^12*B^3 - 1100665640787077376*x^40*y*A^10*B^4 - 5742889576667882496*x^41*y*A^8*B^5 - 8045848338535636992*x^42*y*A^6*B^6 - 4636406057539829760*x^43*y*A^4*B^7 - 970153724430581760*x^44*y*A^2*B^8 - 41844404386004992*x^45*y*B^9 + 2081868962549880*x^34*y*A^19 + 10768360169152224*x^35*y*A^17*B + 260741969978213376*x^36*y*A^15*B^2 + 1105696332226577920*x^37*y*A^13*B^3 + 1634943929953182720*x^38*y*A^11*B^4 - 4860522253044215808*x^39*y*A^9*B^5 - 18185259299370713088*x^40*y*A^7*B^6 - 23340295415514857472*x^41*y*A^5*B^7 - 9455819414325166080*x^42*y*A^3*B^8 - 714994366210375680*x^43*y*A*B^9 + 2538487471297470*x^32*y*A^20 + 4938213117297840*x^33*y*A^18*B + 159674077167432480*x^34*y*A^16*B^2 + 1354621280545958400*x^35*y*A^14*B^3 + 5208812350876391680*x^36*y*A^12*B^4 + 6015450608356902912*x^37*y*A^10*B^5 - 14804059057960218624*x^38*y*A^8*B^6 - 50783304720310665216*x^39*y*A^6*B^7 - 53425567137380106240*x^40*y*A^4*B^8 - 12789734049440071680*x^41*y*A^2*B^9 - 262667338773430272*x^42*y*B^10 + 2426936257526880*x^30*y*A^21 + 11127753499695360*x^31*y*A^19*B + 61668134973403200*x^32*y*A^17*B^2 + 930904307722206720*x^33*y*A^15*B^3 + 6029608774158789120*x^34*y*A^13*B^4 + 20062780231079596032*x^35*y*A^11*B^5 + 19790995374247657472*x^36*y*A^9*B^6 - 36491892793689636864*x^37*y*A^7*B^7 - 122012582934277914624*x^38*y*A^5*B^8 - 90704095535520808960*x^39*y*A^3*B^9 - 10517403941380030464*x^40*y*A*B^10 + 1696580718326448*x^28*y*A^22 + 17013167618521536*x^29*y*A^20*B + 60234577174506240*x^30*y*A^18*B^2 + 381946382350917120*x^31*y*A^16*B^3 + 4159131490833580800*x^32*y*A^14*B^4 + 22433531142090765312*x^33*y*A^12*B^5 + 64979639941811933184*x^34*y*A^10*B^6 + 59086360834020458496*x^35*y*A^8*B^7 - 79232480419818700800*x^36*y*A^6*B^8 - 240363895120741007360*x^37*y*A^4*B^9 - 104485562491597750272*x^38*y*A^2*B^10 - 3902901613781581824*x^39*y*B^11 + 907560369156192*x^26*y*A^23 + 14306349236932224*x^27*y*A^21*B + 99474411797913600*x^28*y*A^19*B^2 + 242477286523169280*x^29*y*A^17*B^3 + 1941894157686067200*x^30*y*A^15*B^4 + 14377136352762839040*x^31*y*A^13*B^5 + 72771756636016779264*x^32*y*A^11*B^6 + 175153360755721666560*x^33*y*A^9*B^7 + 163115052450946744320*x^34*y*A^7*B^8 - 170970271166053220352*x^35*y*A^5*B^9 - 352304775105376944128*x^36*y*A^3*B^10 - 70106489441721778176*x^37*y*A*B^11 + 386288639722308*x^24*y*A^24 + 8228741098923456*x^25*y*A^22*B + 87607101541325184*x^26*y*A^20*B^2 + 438407993230394880*x^27*y*A^18*B^3 + 981788616192111360*x^28*y*A^16*B^4 + 7175172121225371648*x^29*y*A^14*B^5 + 42934712506577633280*x^30*y*A^12*B^6 + 198875673228087263232*x^31*y*A^10*B^7 + 411593760714182492160*x^32*y*A^8*B^8 + 370428770703129968640*x^33*y*A^6*B^9 - 333497117491272351744*x^34*y*A^4*B^10 - 335430537364157497344*x^35*y*A^2*B^11 - 19665775940669014016*x^36*y*B^12 + 120954844413872*x^22*y*A^25 + 3898468576998144*x^23*y*A^23*B + 46675751997102336*x^24*y*A^21*B^2 + 429364204725711360*x^25*y*A^19*B^3 + 1514352362146122240*x^26*y*A^17*B^4 + 4131045401518522368*x^27*y*A^15*B^5 + 19080125103636086784*x^28*y*A^13*B^6 + 118589226305883734016*x^29*y*A^11*B^7 + 444236358785178009600*x^30*y*A^9*B^8 + 884893785796126965760*x^31*y*A^7*B^9 + 600182863343690711040*x^32*y*A^5*B^10 - 449628989590540910592*x^33*y*A^3*B^11 - 172637135404529614848*x^34*y*A*B^12 + 24406973693256*x^20*y*A^26 + 1383103275723488*x^21*y*A^24*B + 21623159441819904*x^22*y*A^22*B^2 + 215726629324905984*x^23*y*A^20*B^3 + 1634866861525551360*x^24*y*A^18*B^4 + 4852173177755882496*x^25*y*A^16*B^5 + 13384163107097395200*x^26*y*A^14*B^6 + 44631917838383382528*x^27*y*A^12*B^7 + 278361275412880687104*x^28*y*A^10*B^8 + 858409417355388518400*x^29*y*A^8*B^9 + 1643202476113924194304*x^30*y*A^6*B^10 + 681239054352453206016*x^31*y*A^4*B^11 - 332181781120467075072*x^32*y*A^2*B^12 - 34201864154615119872*x^33*y*B^13 + 3489467691248*x^18*y*A^27 + 283160582291136*x^19*y*A^25*B + 8459338339144704*x^20*y*A^23*B^2 + 88190815411325440*x^21*y*A^21*B^3 + 832997479741056000*x^22*y*A^19*B^4 + 5035360671800414208*x^23*y*A^17*B^5 + 15002165577455271936*x^24*y*A^15*B^6 + 32089157098943938560*x^25*y*A^13*B^7 + 105912310217463300096*x^26*y*A^11*B^8 + 521444709873766891520*x^27*y*A^9*B^9 + 1532780478042816380928*x^28*y*A^7*B^10 + 2411187739143004225536*x^29*y*A^5*B^11 + 638147167197486645248*x^30*y*A^3*B^12 - 103633925249905459200*x^31*y*A*B^13 + 774955167252*x^16*y*A^28 + 26376200064864*x^17*y*A^26*B + 1916049951984960*x^18*y*A^24*B^2 + 34413088973964800*x^19*y*A^22*B^3 + 310554321428727552*x^20*y*A^20*B^4 + 2570452669693151232*x^21*y*A^18*B^5 + 13750219448169222144*x^22*y*A^16*B^6 + 39753384831263047680*x^23*y*A^14*B^7 + 69167471257133383680*x^24*y*A^12*B^8 + 217488301081293225984*x^25*y*A^10*B^9 + 837335526412577144832*x^26*y*A^8*B^10 + 2414502003790539915264*x^27*y*A^6*B^11 + 2769465444486464143360*x^28*y*A^4*B^12 + 529519463964176547840*x^29*y*A^2*B^13 - 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