curve #486

y² + xy + y = x³ − x² − 682921194634516400005398310342630856170904111380877x + 6821955445671050353247067807452796578532956141061962997054844323393510773429

a-invariants: [1, -1, 1, -682921194634516400005398310342630856170904111380877, 6821955445671050353247067807452796578532956141061962997054844323393510773429]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 6467420206652532321375335216913420856551465130881758563694664862777597598401736912029710
discriminant (Δ): 279228833446562741971527345897062313521266003657925183217334855335599415572366692915757669651945351089113804714819908088282239548453240121026560000000000
Faltings height: 28.0795
naive height: 362.7650
primes of bad reduction: 2, 3, 5, 7, 11, 19, 31, 43, 61, 89, 131, 199, 211, 263, 283, 337, 443, 619, 809, 1873, 3083, 60317, 738937, 3584363, 7317619, 32849761151039335443737
regulator: 7172760148631977039.636466213996417733839847213041880300767434487524233
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:31 UTC
last updated: 2026-09-01 19:28:31 UTC

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

(-9762391056417557724227313, 112064720434715281659675947942120753656)
(10199426602031421175212687, 30291581125967387741227549677016913656)
(-26703193566724898198579163, 77570216542872641214250610195929927756)
(74339631351274555932687, 82287227117269256429427209646133073656)
(8143638333462863185930437, 42433111094730504144586715967253726156)
(6347306706614733462487633, 52373346074850275066976699893352771468)
(-28772647905236080099758063, 51493015757249490161724968609247766156)
(1301108132880556040430486243/169, 98756766959721788446292536288843732084012/2197)
(62105722728783024006723987, 451617001178015208216478957845570761656)
(13853847957299042912492637, 4452601827342210622364850072522422956)
(1201183118016690361530187, 77481450372659130192040955011523736156)
(34512803858690635684452687, 156082643341551822859487225491756073656)
(13557962909925774806526039, 7425476547694768209980691301019144848)
(75356667225628874837100427813/4624, 1431570522933894048835279774541524415662767/314432)
(29822020305213842678468937, 113921753006743501549783126243157344906)
(576001391543572697044040838807/32761, 93735370102407504555104965633533572570803096/5929741)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-4046/2511 on branch u=19/13 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 03:27:02 UTC · history