curve #6473

y² + xy = x³ − x² + 5632957919284911x + 4125454070876803586814349

a-invariants: [1, -1, 0, 5632957919284911, 4125454070876803586814349]
c-invariants: [-270381980125675719, -3564392318454277209573138285]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 3787966790370899764343282737908640664928080965962
discriminant (Δ): -7363807440481029141883341642494397452620189397830128
Faltings height: 8.6315
naive height: 126.8816
primes of bad reduction: 2, 3, 67, 25969, 57993850631, 12256250076377, 170162582965031809
regulator: 8935850023230442.80460126410040034019593300494902312784178558109776003937
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:54 UTC
last updated: 2026-11-04 02:10:54 UTC

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

(8871417898/81, -1794058503190237/729)
(46138117578, -9910404978455125)
(168056104174/729, -82753460014835215/19683)
(689212020422/961, -578484153749781607/29791)
(381753413/16, 132301910670815/64)
(-33328445186/225, 683281400086357/3375)
(-226035677981/1521, -4705632302138708/59319)
(3500531456726/10609, 7073705434791822881/1092727)
(1835477302423/3969, 2569862608286609035/250047)
(-82325875, 1761786935543)
(-37616230, 1964792888627)
(183694958, 3370186011047)
(216038183, 3927414020645)
(308131082, 5925770013083)
(639806723, 16420277864372)
(53686723121, 12439411597515875)
(-1920419926/49, 672539135580605/343)

Commentary

Mestre ten-root construction, roots 0,9,27,31,64,75,78,103,126,137: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 02:10:54 UTC · history