|
%I #16 Jul 22 2024 15:25:14
%S 0,1,1,-1,1,-1,-3,2,3,-5,-11,-5,1,8,31,38,7,-13,-19,53,94,-249,89,36,
%T -41,33,479,-4335,-2357,-149,181,-151,-350,919,5959,18041,-8767,-4544,
%U 1535,989,-493,-2591,-12016,182879,3085709,496035,-48259,-11811,-1466,6627,13751,-55287,-201383,5002782,-124991065
%N Elliptic net associated to y^2 + y = x^3 + x^2 - 2*x, based on the non-torsion generator points P = [0, 0] and Q = [1, 0].
%C The curve y^2 + y = x^3 + x^2 - 2*x is one of the rank-2 elliptic curves with smallest conductor.
%C The signs are defined by the Weierstrass sigma function. In the literature are other variants of sign assignment for this particular net presented.
%H Katherine E. Stange, <a href="https://eprint.iacr.org/2006/392.pdf">The Tate Pairing via Elliptic Nets</a>, Proc. of Pairing Conference 2007, T. Takagi et al. eds., LNCS, Vol. 4575, pp. 329-348, Springer-Verlag, Berlin, 2007.
%F A(n, k) = ws(z1 + z2)/(ws(z1)^(n^2 - k*n)*ws(z1 + z2)^(k*n)*ws(z2)^(k^2 - k*n)), where ws is the Weierstrass sigma function using the lattice parameters of y^2 + y = x^3 + x^2 - 2*x, z1 is the lattice point corresponding to P = [0, 0] and z2 corresponds to Q = [1, 0].
%F A(n*c1, n*c2) divides A((n*k)*c1, (n*k)*c2), where c1, c2 are some integer constants not equaling zero simultaneously and k >= 1.
%F A(n, k) = A(n-1, k)*A(n-3, k)*A(2, 0)^2 - A(1, 0)*A(3, 0)*A(n-2, k)^2)/A(n-4, k), for n > 4.
%F A(n, k) = A(n, k-1)*A(n, k-3)*A(0, 2)^2 - A(0, 1)*A(0, 3)*A(n, k-2)^2)/A(n, k-4), for k > 4.
%F A(n, n) = A(n-1, n-1)*A(n-3, n-3)*A(2, 2)^2 - A(1, 1)*A(3, 3)*A(n-2, n-2)^2)/A(n-4, n-4), for n > 4.
%F |A(n, 0)| = |A178624(n)|.
%e A(n, k) is a square array read by ascending antidiagonals:
%e .
%e --> k*Q
%e n*P | 0, 1, -1, -5, 31, 94
%e | | 1, 1, 3, 8, 53, 479
%e | | -1, 2, 1, -19, 33, 919
%e \ / | -3, -5, -13, -41, -350, -2591
%e |-11, 7, 36, -151, -493, 13751
%e | 38, 89, 181, 989, 6627, 68428
%e .
%e n*P means elliptic point multiplication here. A(n, k)^2 is the denominator of the x coordinate from n*P + k*Q with point multiplication and addition under the elliptic group law for rational numbers.
%o (PARI)
%o T(n, k) = { local(E, z1); local(E, z2); E=ellinit([0,1,1,-2,0]); z1=ellpointtoz(E,[0,0]); z2=ellpointtoz(E, [1,0]);round(ellsigma(E, n*z1+k*z2)/(ellsigma(E, z1)^(n^2-k*n)*ellsigma(E, z1+z2)^(k*n)*ellsigma(E, z2)^(k^2-k*n) )) }
%o A(size) = { my(si = max(0, size-5)); M = matconcat([matrix(5,5,m,k,T(m-1,k-1)),matrix(5, si);matrix(si, 5),matrix(si, si)]);
%o for(k = 1, 5, for(n = 6, size, M[n, k] = (M[n-1, k]*M[n-3, k]*M[3, 1]^2 - M[2, 1]*M[4, 1]*M[n-2, k]^2)/M[n-4, k]));
%o for(k = 1, 5, for(n = 6, size, M[k, n] = (M[k, n-1]*M[k, n-3]*M[1,3]^2 - M[1, 2]*M[1, 4]*M[k, n-2]^2)/M[k, n-4]));
%o for(k = 6, size, for(n = 6, size, M[n, k] = (M[n-1, k]*M[n-3, k]*M[3, 1]^2 - M[2, 1]*M[4, 1]*M[n-2, k]^2)/M[n-4, k])) }; M;
%o sd1(P)=sqrtint(denominator(P[1]));
%o Pnm(n, m, E, P1, P2) = elladd(E, ellmul(E, P1, n), ellmul(E, P2, m));
%o Aunsigned(size) = my(E=ellinit([0,1,1,-2,0]), P=[0,0], Q=[1,0]); matrix(size, size, m, n, sd1(Pnm(m-1, n-1, E, P, Q)));
%Y Cf. A178624.
%K sign,tabl
%O 0,7
%A _Thomas Scheuerle_, Jul 21 2024
|
