curve #3128

y² + xy + y = x³ + x² − 2435109968648812284250327501767338349106208713x + 46251325990521754026689983318486396751949337631251172036297849383406

a-invariants: [1, 1, 1, -2435109968648812284250327501767338349106208713, 46251325990521754026689983318486396751949337631251172036297849383406]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 326885688138289438101160708711925723710232867288025
discriminant (Δ): 7631854496070135847948811569256638078651460559877101792277786672630094212981289373425852593952779386796340056096436718303939252890625
Faltings height: 24.6991
naive height: 325.1326
primes of bad reduction: 3, 5, 7, 19, 31, 37, 41, 53, 73, 83, 97, 109, 157, 271, 359, 431, 449, 457, 797, 1447, 3061, 6148501
regulator: 3402742020973.65707153107829281733749692907823427641493266648
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:35 UTC
last updated: 2026-09-26 20:38:35 UTC

Witness: 11 independent points log in to add more points →

(2904461205393356616749568370/101761, -196259304746224466173105459227884035557/32461759)
(2336758164503567963573146/81, 75891312257818910050131295384379600/729)
(44620034734216376701146, 5141282057754470438558285970038005)
(264949352425176615239434/9, -7518507629236795409756498212876880/27)
(2333005701994781703380313654162084/80787261361, -2591089264917084454628104716727912103034692543278/22962244083898391)
(669051322617160123393807354/23409, -80911793908565648777652413965551928720/3581577)
(28249476777825481619230, 68966919135415511094965346852397)
(252796269802836187596490557241/8162449, -17150151525785729185288845369671070910309941/23320116793)
(246717565079580214509107770/6561, 1489473360080585162901430465465866830032/531441)
(1185630665602514780864786278200639871/9993450982564, -1193519699594630110616373468129293666780154113208316989/31591716971022584488)
(17559874871246370713569321/625, 1785408258544403483197621618853651919/15625)

Commentary

2-isogenous image of D-z2z2old1011121314-056, rational kernel x=-228301316031920956063565/4. Source: https://dujella.github.io/tors/z2z2old1011121314.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:38:35 UTC · history