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A374833
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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].
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0
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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
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OFFSET
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0,7
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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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.
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EXAMPLE
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A(n, k) is a square array read by ascending antidiagonals:
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--> 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
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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.
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PROG
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(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)));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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