curve #3999

y² + xy = x³ − 21907828478303368010635x + 1248028747220975909204449832490225

a-invariants: [1, 0, 0, -21907828478303368010635, 1248028747220975909204449832490225]
c-invariants: [1051575766958561664510481, -1078296837600500549203082497768320121]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 8517544174435024143004460515381044430
discriminant (Δ): 69794403730586320988601187786447125765118962818761998220526720000
Faltings height: 11.5036
naive height: 165.9370
primes of bad reduction: 2, 5, 31, 151, 193, 199, 241, 311, 409, 1031, 1721, 2333, 2437, 2711, 5651
regulator: 6834075770812159613.654171211417756425642445726615359654427006552
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:42:58 UTC
last updated: 2026-10-05 04:42:58 UTC

Witness: 13 independent points log in to add more points →

(12921074909308473707143817902/20095038597001, 1468750225749362607427326272359176790434309/90081014175667635749)
(78400493914408568492590/43734938641, -21879645145112636398300520154589815/9146243983053689)
(79072786865973460/24649, -22211922560113929001340905/3869893)
(584855021779465531270/5773112361, -3626082616125801748432448686235/438646850301141)
(37376842090344422005/401120784, 31829499291409293464076178705/8033647061952)
(37505176948053655024270/421732049281, -480258634120904204700171138386415/273876588391524929)
(75898078623859500190/13111641, 661006090242751209890417208445/47477252061)
(11892053159319008500026708670/137738714250198889, -18826577591764503419657496444702690718805/51119198630112968756117963)
(141743432239032243215620/1580066598049, 4281442736440154084090564138816015/1986154774213779343)
(11265924127192495004104330/130849874515369, 304718550238512990083074514203793455/1496786873135948922347)
(215999690152284618775/2492605476, 68915183167755027836296547015/124445820994776)
(1011717182596484773080730210/11655341519448649, -799124833574161755780053068060253579565/1258309586492914837315643)
(334599681395808207968281510/3812327833890601, 270287827483557393790015350467566964315/235388564204460796564651)

Commentary

Unlisted rational-isogeny neighbor of curve#3635 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 13: 13 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:42:59 UTC · history