curve #483

y² + xy = x³ − 268580253302501973944982309310683755794969010495035395x + 53369999163244032199319178851772476718961242731802220310937173812405654458857025

a-invariants: [1, 0, 0, -268580253302501973944982309310683755794969010495035395, 53369999163244032199319178851772476718961242731802220310937173812405654458857025]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 7063699993905304395318647806097462125358626581250568524161592201139470720695407656860165270
discriminant (Δ): 9454256943012350934702661920675584939579298706938543859673337210713612369846625557656301263594391544316105011524245447139937687162888903902605143572582400000000
Faltings height: 29.5489
naive height: 380.6886
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 31, 41, 43, 53, 73, 131, 149, 179, 197, 277, 463, 661, 2459, 3491, 31727, 19244683, 54275107, 196645627, 116472563371, 117763093086091
regulator: 52444020861068508565.62053828799327271421976996051831081831962818096236
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:19 UTC
last updated: 2026-09-02 05:26:49 UTC

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

(-139291217851561028139805650, 9385005886345300525380242203756104645705)
(269225166767042949115137390, 758594411532241394966630854354750216905)
(335438250799717991736200526, 1010457813249357395013510626814990464937)
(54724115460483571535545619160/169, 1306266520584792583115984193326735706705285/2197)
(14173911517019862289967190, 7040313484018062416268931623356609017905)
(385271275362245872147090350, 2661030218686936270326513730505618823945)
(512440124316043215885069690, 7092450510303323830685670818890610130405)
(478365962914660898584516590, 5861462427338199001682920784359342600905)
(1881516690433737539486797860, 78795929586005385103779968242488710794185)
(-150361029184488335995007666195581410/885598081, 255620391045058348891571455100867773275608038242922695/26354513292479)
(69571280300618504723193047910/289, 8020540069312968700760437486032303185792265/4913)
(314236939272097380967332810, 37988252712091540271952609560328881205)
(8977851054808501036809990060630/1369, 26818815392208276587716370493548495691173122965/50653)
(17759702643194285056144960965/16, 2144537929565495132650751292817717157311845/64)
(2863077463497444262604620802190/7921, 1321548759762463499879509949187436379090580945/704969)
(-137399075376770637314330310, 9363695230748150021997939119400318762405)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=6345/5704 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 03:27:00 UTC · history