curve #3108

y² + xy + y = x³ − x² − 1060782949216038329262375848436468363024926914782774467x + 420519802995751124724111740218478048655771772712848510327017985388378553526801859

a-invariants: [1, -1, 1, -1060782949216038329262375848436468363024926914782774467, 420519802995751124724111740218478048655771772712848510327017985388378553526801859]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2603348955675569045873307484526694064917261974747900874693810
discriminant (Δ): 512528915611657524637710374768288075919700064229683424846841011607507715694302155250803756901555862053112766475767218160083261220971276882971354228178576000000
Faltings height: 29.6667
naive height: 384.8094
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 37, 67, 71, 83, 97, 127, 149, 191, 193, 683, 947, 1481, 1951, 3457, 14008947836626425239
regulator: 331838289473254.499228422681649951649090171842401136490151528
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:40 UTC
last updated: 2026-09-26 20:37:40 UTC

Witness: 11 independent points log in to add more points →

(692624334898482291199048909, 4250568232561562704733904331849250374318)
(55702708081130513111184565109093863/41822089, 10012203505930283299696259303333708680591000477985558/270463449563)
(40560081964395424286100793617117105888087/68032489808761, -30078748344396018962557790774454783294489794752278584452294/561144289823316113741)
(1559893118134906504218329826236204647/2619290041, 878728641659205317442419488367932927605969043746074/134052645008339)
(70019784835122227017487873391934723/117440569, -70031055363570511824411389237782305005136286870902/1272703446253)
(692340483050030639274360157302205886547/1151391234841, -345592066525454767124899692434134347159109856549477867126/1235476185330203389)
(1033510088965345548624155450746418960163/150228333649, 32872782689454871815787627899982428901047108564526161644522/58227450524016857)
(153279794220595517662364047036137/118336, -1421689830259902262808225917192080461333868022831/40707584)
(275130737257725277715879172725263971751/366650837289, 1522883401322519725154430602121015600491643271479645271334/222013315042723413)
(84401229115329331484935298674651128342074463/118073169798066289, -212767298180201410638219461980841470492085806116130025249088514942/40572038948623379144156663)
(5094421144436612630554906383576863299802372296192331757413872557/8537045762208871153966935693260991961, -2011407126029310492663560264599158255249840998956499305404474643372276229907891453585149308214/24943730990103314694843612009754080320811192815217736291)

Commentary

2-isogenous image of D-z4old9101112-009, rational kernel x=-296874251695587001057277733. Source: https://dujella.github.io/tors/z4old9101112.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:40 UTC · history