curve #335

y² + xy + y = x³ − x² − 109648466330311563423519471692x + 14831224909223981171042503872168327240436559

a-invariants: [1, -1, 1, -109648466330311563423519471692, 14831224909223981171042503872168327240436559]
c-invariants: [5263126383854955044328934641201, -12814178321569496047711995998255735255531301801]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 26131480673084455535055442121869459110183352891317952147665051014474190
discriminant (Δ): -10655056235250006421362330734062242646577351173136784942681626012228895953862213427200000
Faltings height: 15.6402
naive height: 212.3338
primes of bad reduction: 2, 3, 5, 11, 17, 71, 2223291744473, 81117610011058082651461, 121257874715182550773823852011
regulator: 10713426441429307164.08685871875394308560929749321424630343268659017739686607
submitted by: Andrii Prokhorenkov
submitted at: 2026-08-24 19:18:32 UTC
last updated: 2026-08-24 19:18:32 UTC

Witness: 19 independent points log in to add more points →

(221608685343698629923/41209, -3292881277111888095288314573803/8365427)
(490362676990797693/169, -341242384963643848709435783/2197)
(233965481556897073/169, -110209435812361409670530723/2197)
(1115362331312277, -35778222258987515923019)
(1407589819737983959881/1530169, -49805952524110440557323347536579/1892819053)
(-5188162901069531283/72361, 91969830766048396031072701049/19465109)
(59467225364789816623/368449, 258743453724598266929113530363/223648543)
(210266770728117, 1035439221889299277141)
(436374416467848621/1849, 115182735830651444642458351/79507)
(462536637767410249437/1394761, 6366728134940215445202805472681/1647212741)
(778679660054993691/1225, 609331599371070275349844559/42875)
(161234648327618978187/28561, 2043888117571704073572039203269/4826809)
(48416703022599357, -10653264032295390235339259)
(6593806095211503381/139129, 161968109309531435841565562881/51895117)
(148833567463192797/16, -57382571347653779996094101/64)
(117559475102469, 1888308117378624822037)
(35158120425305467/196, 2652541571506819465114179/2744)
(23865502012886843668533/113569, 3686848755353665471988237369207653/38272753)
(161019268615172093628027/3818116, 64610548578070432026379876694050729/7460598664)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=5015/22. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by Andrii Prokhorenkov at 2026-08-24 19:18:32 UTC · history