A374833 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].

0, 1, 1, -1, 1, -1, -3, 2, 3, -5, -11, -5, 1, 8, 31, 38, 7, -13, -19, 53, 94, -249, 89, 36, -41, 33, 479, -4335, -2357, -149, 181, -151, -350, 919, 5959, 18041, -8767, -4544, 1535, 989, -493, -2591, -12016, 182879, 3085709, 496035, -48259, -11811, -1466, 6627, 13751, -55287, -201383, 5002782, -124991065

Offset: 0

Author

Thomas Scheuerle, Jul 21 2024

Keywords

• sign
• tabl

Comments

The curve y^2 + y = x^3 + x^2 - 2*x is one of the rank-2 elliptic curves with smallest conductor.

The signs are defined by the Weierstrass sigma function. In the literature are other variants of sign assignment for this particular net presented.

Examples

			A(n, k) is a square array read by ascending antidiagonals:
.
      --> k*Q
 n*P |  0,  1,  -1,   -5,   31,    94
  |  |  1,  1,   3,    8,   53,   479
  |  | -1,  2,   1,  -19,   33,   919
 \ / | -3, -5, -13,  -41, -350, -2591
     |-11,  7,  36, -151, -493, 13751
     | 38, 89, 181,  989, 6627, 68428
.
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.
		

Links

• Katherine E. Stange, The Tate Pairing via Elliptic Nets, Proc. of Pairing Conference 2007, T. Takagi et al. eds., LNCS, Vol. 4575, pp. 329-348, Springer-Verlag, Berlin, 2007.

Crossrefs

Cf. A178624.

Programs

PARI

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) )) }
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)]);
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]));
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]));
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;
sd1(P)=sqrtint(denominator(P[1]));
Pnm(n, m, E, P1, P2) = elladd(E, ellmul(E, P1, n), ellmul(E, P2, m));
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)));

Formula

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].

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.

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.

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.

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.

|A(n, 0)| = |A178624(n)|.