{{{id=1| x=var('x') K.=NumberField(x^2-x-1) /// }}} {{{id=2| %cython include 'sage/libs/pari/decl.pxi' from sage.libs.pari.gen import pari from libc.stdint cimport uint8_t, uint_fast8_t, uint32_t, uint_fast32_t, uint_fast64_t cdef extern from "pari/pari.h": cdef void NEXT_PRIME_VIADIFF(uint32_t, uint_fast8_t *) cdef uint_fast32_t[32] shiftTab shiftTab[ 0] = 0x00000001u shiftTab[ 1] = 0x00000002u shiftTab[ 2] = 0x00000004u shiftTab[ 3] = 0x00000008u shiftTab[ 4] = 0x00000010u shiftTab[ 5] = 0x00000020u shiftTab[ 6] = 0x00000040u shiftTab[ 7] = 0x00000080u shiftTab[ 8] = 0x00000100u shiftTab[ 9] = 0x00000200u shiftTab[10] = 0x00000400u shiftTab[11] = 0x00000800u shiftTab[12] = 0x00001000u shiftTab[13] = 0x00002000u shiftTab[14] = 0x00004000u shiftTab[15] = 0x00008000u shiftTab[16] = 0x00010000u shiftTab[17] = 0x00020000u shiftTab[18] = 0x00040000u shiftTab[19] = 0x00080000u shiftTab[20] = 0x00100000u shiftTab[21] = 0x00200000u shiftTab[22] = 0x00400000u shiftTab[23] = 0x00800000u shiftTab[24] = 0x01000000u shiftTab[25] = 0x02000000u shiftTab[26] = 0x04000000u shiftTab[27] = 0x08000000u shiftTab[28] = 0x10000000u shiftTab[29] = 0x20000000u shiftTab[30] = 0x40000000u shiftTab[31] = 0x80000000u cdef uint_fast8_t[256] twoDiv cdef uint_fast32_t tempItr, tempVar twoDiv[0] = 8u for tempItr in range(1,255u): tempVar = tempItr while not tempVar&1u: twoDiv[tempItr] += 1u tempVar >>= 1u cdef uint_fast32_t exp_mod(uint_fast64_t b, uint_fast32_t e, uint_fast32_t p): cdef uint_fast64_t q if e&1u: q = b else: q = 1ull e >>= 1u while e: b *= b if b > 4294967295ull: b %= p if e&1u: q *= b if q > 4294967295ull: q %= p e >>= 1u if q > 4294967295ull: q %= p return q cdef uint_fast32_t non_residue(uint_fast32_t p): cdef uint8_t *pariPrimePtr = diffptr cdef uint32_t pariP = 0u NEXT_PRIME_VIADIFF(pariP, pariPrimePtr) while True: NEXT_PRIME_VIADIFF(pariP, pariPrimePtr) if exp_mod(p,(pariP-1u)>>1u,pariP)%pariP > 1u: return pariP cdef uint_fast32_t sqrt5_mod(uint_fast32_t p): if p&3u == 3u: return exp_mod(5ull, (p+1u)>>2u, p)%p cdef uint_fast64_t q, z if p&7u == 5u: q = exp_mod(10ull, (p-5u)>>3u, p) z = q*q if z > 1844674407370955161ull: z %= p z *= 10ull z -= 1ull if z > 4294967295ull: z %= p q *= 5ull if q > 4294967295ull: q %= p return q*z%p cdef uint_fast32_t d, dp if p&15u == 9u: q = exp_mod(10ull, (p-9u)>>4u, p) if q > 4294967295ull: q %= p z = q*q if z > 1844674407370955161ull: z %= p z *= 10ull if z > 4294967295ull: z %= p q *= 5ull if q > 4294967295ull: q %= p if z*z%p == 1ull: d = non_residue(p) dp = exp_mod(d, (p-1u)>>3u, p) q *= dp if q > 4294967295ull: q %= p dp *= dp if dp > 4294967295ull: dp %= p z *= dp if z > 4294967295ull: z %= p z -= 1ull return q*z%p p -= 1u cdef uint_fast8_t r = 0u q = 8ull while q == 8ull: q = twoDiv[(p>>r)&0xFFu] r += q q = p>>r p += 1u r -= 2u cdef uint_fast64_t v = exp_mod(non_residue(p), q, p) d = exp_mod(5ull, (q-1u)>>1u, p) cdef uint_fast64_t res = 5ull*d if res > 4294967295ull: res %= p d *= d if d > 3689348814741910323ull: d %= p d *= 5ull d %= p cdef uint_fast8_t m while not d == 1u: m = r dp = d*d%p while not dp == 1u: dp *= dp dp %= p m -= 1u z = exp_mod(v, shiftTab[m], p) res *= z if res > 4294967295ull: res %= p z *= z if z > 4294967295ull: z %= p d *= z d %= p return res%p def get_primes(p): q = sqrt5_mod(p)-1u w = ((p+1u)>>1u)*q%p q = p-1u-w if w < q: return (w,q) return (q,w) /// }}} {{{id=3| def split_prime_combine(p, firstCurves, secondCurves): ret = [] r1,r2 = get_primes(p) if len(firstCurves) < len(secondCurves): firstCurves, secondCurves = secondCurves, firstCurves r1,r2 = r2,r1 for E in firstCurves: liftE = [E[0],0,E[1],0] for i in range(p): for j in range(p): if ((liftE[0]-r2*liftE[1])%p,(liftE[2]-r2*liftE[3])%p) in secondCurves: ret.append(list(liftE)) liftE[2] = (liftE[2]+r1)%p liftE[3] += 1 liftE[3] = 0 liftE[0] = (liftE[0]+r1)%p liftE[1] += 1 return ret def integral_combine(n, first, second, firstCurves, secondCurves): ret = [] for E in firstCurves: for i in range(second): for j in range(second): for k in range(second): for l in range(second): if [A%second for A in E] in secondCurves: ret.append(list(E)) E[3] = (E[3]+first)%n E[2] = (E[2]+first)%n E[1] = (E[1]+first)%n E[0] = (E[0]+first)%n return ret def curves_in_GF(p): ret = {} K = GF(p) for i in range(p): for j in range(p): try: E = EllipticCurve(K,[i,j]) ap = p-len(E.points())+1 if ret.has_key(ap): ret[ap].append(E) else: ret[ap] = [E] except ArithmeticError: pass return ret def various_lifts(A,B,p): q=(p+1)/2 if p%3==1: v = (2*p+1)/3 if p%3==2: v = (p+1)/3 ret = [] for s in [0,q,q*a,q+q*a]: for r in [v*s^2,v*(s^2+1),v*(s^2-1),v*(s^2+a),v*(s^2-a),v*(s^2+a+1),v*(s^2+a-1),v*(s^2-a+1),v*(s^2-a-1)]: for t in [0,q,q*a,q+q*a]: a1 = K(2*s)[0]%p+K(2*s)[1]%p*a a2 = K(3*r-s^2)[0]%p+K(3*r-s^2)[1]%p*a if a2[0]>1: comp0=a2[0]-p else: comp0 = a2[0] if a2[1]>1: comp1 = a2[1]-p else: comp1 = a2[1] a2 = comp0+comp1*a a3 = K(2*t)[0]%p+K(2*t)[1]%p*a a4 = K(A+3*r^2-2*s*t)[0]%p+K(A+3*r^2-2*s*t)[1]%p*a a6 = K(B+r*A+r^3-t^2)[0]%p+K(B+r*A+r^3-t^2)[1]%p*a for c in [a4,a4-p,a4-p*a,a4-p-p*a]: for d in [a6,a6-p,a6-p*a,a6-p-p*a]: #print (a1,a2,a3,c,d) ret.append((a1,a2,a3,c,d)) return ret def quick_disc(ainv): a1 = ainv[0] a2 = ainv[1] a3 = ainv[2] a4 = ainv[3] a6 = ainv[4] return -(a1^2+4*a2)^2*((a1^2+4*a2)*a6-a1*a3*a4+a2*a3^2-a4^2)-8*(a1*a3+2*a4)^3-27*(a3^2+4*a6)^2+9*(a1^2+4*a2)*(a1*a3+2*a4)*(a3^2+4*a6) def ap(E,p): return E.change_ring(p.residue_field()).trace_of_frobenius() def check_aps_against(E,aps): ret = True for i,p in enumerate([11,19,29,31,41,59,61,71,79,89]): r1 = K.primes_above(p)[0] r2 = K.primes_above(p)[1] if type(aps[2*i]) != str: if ap(E,r1) != aps[2*i] or ap(E,r2) != aps[2*i+1]: ret = False return ret /// }}} {{{id=4| aps = [0,0,-8,4,-10,-2,0,4,10,2,0,4,14,-2,-16,8,-12,-4,'?','?'] di={K(3*a-2): aps[0], K(3*a-1): aps[1], K(-4*a+1): aps[2], K(-4*a+3): aps[3], K(-a+6): aps[4], K(a+5): aps[5], K(5*a-2): aps[6], K(5*a-3): aps[7], K(a-7): aps[8], K(a+6): aps[9], K(7*a-2): aps[10], K(7*a-5): aps[11], K(7*a-3): aps[12], K(7*a-4): aps[13], K(a-9): aps[14], K(a+8): aps[15], K(-8*a+5): aps[16], K(-8*a+3): aps[17], K(a-10): aps[18], K(a+9): aps[19]} N=445 cond = -19*a-7 found = False for p in (11,19,29,31,41,59,61,71,79,89): print 'CURRENT PRIME: ',p r1 = K.primes_above(p)[0] r2 = K.primes_above(p)[1] ap1 = di[r1.gens_reduced()[0]] ap2 = di[r2.gens_reduced()[0]] if type(ap1) == str: continue c = r1.integral_basis()[1][0] d = r2.integral_basis()[1][0] if c>d: r1,r2 = r2,r1 ap1,ap2 = ap2,ap1 modp = curves_in_GF(p) firstp = [] for E in modp[ap1]: firstp.append((int(E.a4()),int(E.a6()))) secondp = [] for E in modp[ap2]: secondp.append((int(E.a4()),int(E.a6()))) big_list_newp = split_prime_combine(p,firstp,secondp) print 'Length of big_list_new%s:'%p,len(big_list_newp) current = 1 for e in big_list_newp: print current current += 1 A = e[0]+e[1]*a B = e[2]+e[3]*a for ainv in various_lifts(A,B,p): D = quick_disc(ainv).norm() if D%N==0: E = EllipticCurve(K,list(ainv)) if E.conductor() == cond: print E, 'tentative' if check_aps_against(E,aps): print E, 'a_ps match' found = True break if found: break if found: break /// WARNING: Output truncated! CURRENT PRIME: 11 Length of big_list_new11: 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 ... 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 }}} {{{id=5| E=EllipticCurve([a+1, -a-1, 0, -a-3, 2*a-9]) /// }}} {{{id=6| E.conductor() /// Fractional ideal (18*a - 12) }}} {{{id=7| /// }}}