curve #3969

y² = x³ − 1999716720835501016846757300x + 24310580713109005131956459258980377572000

a-invariants: [0, 0, 0, -1999716720835501016846757300, 24310580713109005131956459258980377572000]
c-invariants: [95986402600104048808644350400, -21004341736126180434010380799759046222208000]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1213220854870659463310025004843200
discriminant (Δ): 256468599864180747848196024749326478809810189753372342845504131166600954177600000000
Faltings height: 14.7109
naive height: 200.2020
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 37, 41, 79, 83, 127, 131, 163, 191, 433, 853, 883
regulator: 148951370174809.0572729585216845196351676563743242540027773
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:36:31 UTC
last updated: 2026-10-05 04:36:31 UTC

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

(8451368116133006/80089, -3518551278181779695130150304/22665187)
(2084528927425108585/1478656, 263610242546067720184894291125/1798045696)
(169915083679630578936910/4644286201, -3470962146189557391874319104980000/316503460311949)
(50965325423068585/7056, -59987693666590690291152875/592704)
(115145766418757649849401/578979844, 38133935845947316814398524435750451/13931413006328)
(15065813672843146121/309136, 35478079622315680931779219931/171879616)
(1000563484699084724387785/84200950276, -1152712971386752587920432335030894875/24432926545388024)
(7269925345205522104965826663390/93647520030305329, 16634303831241475909895807243240241415125996000/28657903856542961111207767)
(25461287055987325882767310/679010608441, -25335248244384743193790458617903028000/559519000578161261)
(23428695240547411481087710/11107262881, -113377035596240647397521938206463462000/1170605542291471)

Commentary

Unlisted rational-isogeny neighbor of curve#3697 (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 04:36:31 UTC · history