curve #6068

y² + xy + y = x³ − x² − 90980855766592156488x + 334108410959987925402723235467

a-invariants: [1, -1, 1, -90980855766592156488, 334108410959987925402723235467]
c-invariants: [4367081076796423511409, -288669667049777702702368969642377]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 57135593756391946893120530343475250404125239830
discriminant (Δ): -25369881019170372075833759456482477353310990825640948326400
Faltings height: 10.1840
naive height: 149.4857
primes of bad reduction: 2, 5, 7, 11, 31, 293, 76862187676199, 106285486374011684312128487
regulator: 167826353034492610530.277556700124212037412471497187048251433188681255359
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:13:50 UTC
last updated: 2026-10-28 21:13:50 UTC

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

(-3619860877, 784867398713483)
(-10364502244571/5625, 296963239642294851067/421875)
(19562980575013/3481, 3557107364367974507/205379)
(123819409370077/29584, 830468493037170173719/5088448)
(39652614444437/4489, 140967216199995448579/300763)
(1059648043495653/196249, 1448054615810995565259/86938307)
(4638754457254621/848241, 8269152424364201032129/781229961)
(84838095072161453/16313521, 2644697500874108068512607/65890311319)
(158203038360043637/19123129, 32131117672739190492885301/83625443117)
(169253655488210287/59644729, 144776187379784714079897051/460636242067)
(362550067338225553/69772609, 23701010295547676244032605/582810602977)
(471295738007895397/6661561, 320748534157242270105102565/17193488941)
(629257476171379053/92179201, 156191006424683205297851993/885012508801)
(908250285555910977/230827249, 674932437787500307262913701/3506958394057)
(8114696529793821003/1457254276, 682523873804229440555897947/55629224732024)
(2707740550120852917/463670089, 441386888713843107260594941/9984208026437)
(10316307255344053517/1903838689, 1220739944610648814388457161/83070193517137)

Commentary

Mestre ten-root construction, roots 0,20,21,26,35,60,75,90,106,131: 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 2026-10-28 21:13:50 UTC · history