curve #489

y² + xy = x³ − 521606582741693786416031311567510247592163054497538x + 4566163127051356743538072825743951320192950093385811863916905842981272810692

a-invariants: [1, 0, 0, -521606582741693786416031311567510247592163054497538, 4566163127051356743538072825743951320192950093385811863916905842981272810692]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 13742738586193407312966995954537545897263198361682836147475206656230039988267492498287005950
discriminant (Δ): 75445254481781595267746905315098659307321544309204814184568588269404249343994019608220476659259323537058499274987211126964564045512340886932530176000000
Faltings height: 27.9914
naive height: 361.9566
primes of bad reduction: 2, 3, 5, 17, 19, 23, 29, 43, 167, 269, 397, 499, 661, 701, 733, 853, 1399, 1699, 5807, 7001, 13381, 32887, 100787, 101089, 194863, 40236799, 53264824471
regulator: 24344958461571653650.09786581344328485485175124500312205662062824277496
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:43 UTC
last updated: 2026-09-02 05:25:52 UTC

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

(-17219631410126251154144668, 91881146379649528412098378937192884334)
(8984833943268959740083578, 24595435600483527448824758957291636612)
(8558463623378774295412772, 26998083487837782855289535816046852014)
(17162864741912187603179132, 25873939037360593175299474031314422134)
(27239229787567201070738372, 102804895311338736914281503410138601614)
(-26188498992214911046038148, 16285037725366346492747633281486061174)
(12452827844565546507592532, 1335118799301180009064346649153334334)
(29716951865008537640344652, 123727778757298476488726777887178592974)
(115824909915473128940098376, 1223923916853011780477540012245429585994)
(1165123299182324293094557132/81, 4578179678176510757981597713779120136606/729)
(7854011508927619753250966, 30885880669459248216865223435832159080)
(-1137354004910603596212884105788/43681, 203112635868112617474730918150756212315242446/9129329)
(62529958279512719520792163286/6241, 9169052650586973837917455100753393827824920/493039)
(886366473263900732534372, 64066566239564254359849993708666381614)
(176016648310258585729161929081332/8696601, 1229981036185603252206440254761801730392495109766/25646276349)
(73055204321308695045930087991892/175561, 623498543912818758241380349853896745479422689626/73560059)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-4182/133 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