curve #490

y² + xy = x³ − 6821191145727370853408643751670739398156863362596405x + 215861958639749379822107278208513879969118249879447334974949728266795324738721

a-invariants: [1, 0, 0, -6821191145727370853408643751670739398156863362596405, 215861958639749379822107278208513879969118249879447334974949728266795324738721]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 3752162734444495288724086395434552280850923039425413583723536398069902141272495003060217447262
discriminant (Δ): 182733193787408683303404850123569749726008492456090345250402562272304830565755456514970303298675275064225076199599883001104915141243888605882861929772613632
Faltings height: 28.6374
naive height: 369.6692
primes of bad reduction: 2, 3, 59, 67, 97, 103, 149, 157, 233, 349, 359, 521, 593, 643, 3943, 4513, 8059, 15461, 53791, 398423, 429389, 434267, 1038119, 3154201, 3622831, 18820031
regulator: 212781497662551853783.3199860734366452299020061886728527888613025117248
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:47 UTC
last updated: 2026-09-01 19:28:47 UTC

Witness: 16 independent points log in to add more points →

(-58066367208452794627045190, 645105575813064804351456766765542947359)
(108683139288816572978211034, 870793559991941262073246104563922624871)
(30949119552185734928872530, 185463340527999753493717028540989346199)
(63090319639309052811707626, 191402213703132948879952318073545932943)
(101728736553943252869515218, 758099349434651159647851028003947664279)
(31745852837834357527311844, 176948785149520450874910610524347373991)
(211493935084280025054400338, 2869368991194281157414919217632028559639)
(298422002992759285781301618, 4975585028913485900020447684384074270903)
(14922158072730348181991823904/289, 178237958308734550519170143408413123959359/4913)
(88330807598635715414633986, 550024440160711143669464793495515883271)
(3522490673748984497389209195344921266/47586368449, 3543734140026335294549902056703046765943268161902522665/10380633172570207)
(8722334662699927308263848019337730/260467321, 664547308597788961072466150920187491933833618872981/4203682093619)
(2417584218570072504032632954595233042978/86647446188521, 175343748081669353533522459403869103990295753088139615597443/806554373595446376181)
(25587556692707521861944308692473394/507826225, 106851636079941328203243615872303569551697264656977/11443863980375)
(955585254184374261426650705824652434/22493100529, 176910200462967277758193631337142573969165104198848719/3373447738037833)
(-52701156310256007821094445956221/562500, 76086575652733017393330989083048679632056493417/421875000)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-9956/75 on branch u=22/17 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).

last edited by Matthias Breddin at 2026-09-02 03:27:04 UTC · history