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curve #6518

y2 + xy = x3 − x2 − 3903048542455789x + 81300771720576338624569
a-invariants
[1, -1, 0, -3903048542455789, 81300771720576338624569]
c-invariants
[187346330037877881, -70243865923519471401177165]
rank (lower bound)
≥ 17
torsion subgroup
trivial
conductor (N)
15320733234434998305899165043884892747847538306
discriminant (Δ)
949885460534969894965748232720863350366547374972
Faltings height
7.9314
naive height
119.3152
primes of bad reduction
2, 31, 1472606987, 8625442757, 29953695619, 649485103870003
regulator
808186593358890.522626701110228905118722332701620105883170842334286145529
submitted by
Michael Rubinstein
submitted at
last updated

Witness: 17 independent points log in to add more points →

Commentary

Mestre ten-root construction, roots 0,2,63,81,103,104,109,141,162,165: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 17: the 17 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at · history

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