curve #3967

y² + xy = x³ − 1804166658196351158549486883338x − 296730704439266427997445156213824676516703708

a-invariants: [1, 0, 0, -1804166658196351158549486883338, -296730704439266427997445156213824676516703708]
c-invariants: [86599999593424855610375370400225, 256375328635526063889793224831461104947376403375]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 749928839103528874724228030523150
discriminant (Δ): 337808783402559017824865234546494336353577625346804136811700506123270218823096293689600000000
Faltings height: 16.4465
naive height: 220.6166
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 41, 47, 53, 71, 151, 179, 449, 4157, 10357, 44893
regulator: 470333555414964.4964860333532486060450867149419614307160244
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:36:03 UTC
last updated: 2026-10-05 04:36:03 UTC

Witness: 10 independent points log in to add more points →

(-29874286575994284/121, -15387816062028928552242606/1331)
(5549960369431476, 400799733635999275643430)
(-12455555654165326195938569598924/23004420154870561, 79670783933251322390347638965965197406541557830/3489128277087560272507441)
(350757137307823887683586838068/78835770860401, 197675004309685253828902405950607627114303530/699978946516728319351)
(-38757311934952937326641842/77458125969, -472665814468916151466568723282807077352/21557603412810297)
(-6355362920616297309223548/14615117449, -35583003718873875871318880316988517826/1766865393761957)
(2698276073385246069852/961, 140161754000766450855423459292794/29791)
(-2653031465383229961352268/13712175801, -10782796800579185102594328580302734734/1605682074121299)
(873229762539226811870443678644/114940334166529, -802869704356977885563339751444805959264612390/1232277966227043239167)
(441577460060214738359364/141300769, 263362704734592563402542722534151290/1679642241103)

Commentary

Unlisted rational-isogeny neighbor of curve#3696 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 10: 10 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 04:36:03 UTC · history