curve #6179

y² + xy = x³ − 22838565855447084625x + 41855387033877832055344377641

a-invariants: [1, 0, 0, -22838565855447084625, 41855387033877832055344377641]
c-invariants: [1096251161061460062001, -36163054398914823637409732374825]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 5240227668680445150085901505663732024266517078
discriminant (Δ): 5596953719326874216969573620134104832085337340321128251392
Faltings height: 9.9349
naive height: 145.3385
primes of bad reduction: 2, 3, 23, 487, 3163, 24651475359537885220952165832338600251
regulator: 2528576445182533080.36221353485738014041907495305732707968510402851183037
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:40:40 UTC
last updated: 2026-10-30 14:40:40 UTC

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

(-5450976316, 66201146520305)
(-2002216798, 282058647983699)
(2082945950, 57627866524271)
(16834437551/4, 1139283378733873/8)
(9256528106, 789665905186043)
(-529561885679488/116281, 8990743940732329366729/39651821)
(-1202601227578/961, 7795702536882810089/29791)
(-184344194950/49, 93639861588933197/343)
(-120577729726/169, 528135996297425567/2197)
(3762152016026/841, 4168176820700651527/24389)
(5221339598378/1681, 2048684017621989379/68921)
(5746777173386/2209, 713321979772332373/103823)
(8714733556538/3481, 3948665190150977617/205379)
(9681866778836/9409, 127264369976487023585/912673)
(12648541632674/4225, 4865815900687621963/274625)
(183386151643250/57121, 550080130906231407949/13651919)
(1874067773429018/635209, 6302752470645430525447/506261573)

Commentary

Mestre ten-root construction, roots 0,13,14,15,20,31,39,72,95,111: 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 2026-10-30 14:40:40 UTC · history