curve #6174

y² = x³ − x² − 2569048307069387108x + 1622434557263521518889966212

a-invariants: [0, -1, 0, -2569048307069387108, 1622434557263521518889966212]
c-invariants: [123314318739330581200, -1401783456735796679884947320000]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 2713126118888294398313627472411615188672713800
discriminant (Δ): -51983442175377342905801136098857098782665422953724000000
Faltings height: 9.4717
naive height: 138.8306
primes of bad reduction: 2, 3, 5, 7, 19, 327421, 764809, 126574813501, 1072655525555227279
regulator: 1038166723505131681954.32302077716042291783391941401419168053820873023929
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:40:03 UTC
last updated: 2026-10-30 14:40:03 UTC

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

(539015088, 19856553758706)
(2160203278, 78443033391184)
(2317144098, 90059365712904)
(4356332386612, 9092461774151905450)
(-334118217987304/383161, 13415880900751524795102/237176659)
(-25368208261328/13689, 6980589274596676298/1601613)
(-21325033455644/17161, 120979850621639773022/2248091)
(-7669363734992/4489, 9627786623315340550/300763)
(-3489601746542/2209, 4329052538828696000/103823)
(-189844631687/729, 938571586201453400/19683)
(264964854712/361, 78870788237346350/6859)
(15153992812190782/18139081, 594686404016327112190400/77254345979)
(160597449920676082/30151081, 61736964830961835829099000/165559585771)
(204473406405697522/750705201, 631591407349942867453838800/20568571802199)
(209463182142946968/195077089, 27501801502091533088363850/2724641702063)
(346196175348408302/3517557481, 7723394440820274493718359800/208622816640629)
(7126339232138670172/8774256241, 6956736883002869111597895650/821893356350711)

Commentary

Mestre ten-root construction, roots 0,10,47,60,72,88,96,98,99,120: 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 14:40:03 UTC · history