curve #4075

y² + xy = x³ − 49347500834819830195051709537338778559642879140914705x + 1486649563151120803901724238073416572624282461514779242391877140543438386551977

a-invariants: [1, 0, 0, -49347500834819830195051709537338778559642879140914705, 1486649563151120803901724238073416572624282461514779242391877140543438386551977]
c-invariants: [2368680040071351849362482057792261370862858198763905841, -1284465222562568374571089745248451978854407820792492352114973905723818064126766889]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 428177454037705020017159567275865585425893230857404351193992550565248483845677250700590
discriminant (Δ): 6736095039739420062470085798005137381134493310669032384139694282023504945415105752565863554612381828427745004974944993311676855401031441643048627240960000000000
Faltings height: 29.3607
naive height: 375.6058
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 167, 227, 229, 277, 367, 547, 617, 733, 883, 2269, 2371, 3943, 4973, 6317, 13309, 25301, 37747, 88873, 93889, 5302471
regulator: 23076576394292139916487.5937315055868522134506693065918998324606354639078552286
submitted by: wgxli
submitted at: 2026-10-09 04:48:49 UTC
last updated: 2026-10-09 04:48:49 UTC

Witness: 20 independent points log in to add more points →

(16084332190533571228936994, 834918604443215905593011217016068201803)
(17863574096070015400104794, 781554362833523873986254626288468252603)
(24633990588292073512545434, 534763894226694569895985988938261031723)
(26490459898742122425861194, 444973150661422917318379945287883755803)
(29903310351195737163680294, 194256812173553879797672483583058598103)
(4910798254575019929705674, 1115541290449364201473474379220727629083)
(6550551954051100133454794, 1078738743881788571748271348869555992603)
(213321513084075874528562834, 816816556255212079582788068502996873923)
(-14372781937251186623052466, 1480858314501227340129612390943511845523)
(217246123360836835272988874, 1009563206739781247922018140835904288283)
(221486211251624962769538734, 1192525701521274748454195695580284834643)
(236273993614606248678849794, 1737011422813628565948494049912135887603)
(314082170847709165101342464, 4119579455081175065039836147896935584013)
(327652060270781076679133684, 4526942907179408270348286206028364221473)
(344631489063443714292556718, 5041044170555578716711549899236598623571)
(397665293623431008531919794, 6689449184486618444007826233957302357603)
(-160041603040464153114668086, 2298936005726739437614055365384156829723)
(-203859332202535337857210906, 1753420689753753619464851705536682474903)
(534438318888738116114894954, 11303186887984723577087525931404506473563)
(4242571316047219455017193134/361, 6537052827280404538739509522913818766501477/6859)

Commentary

2-isogenous to curve #4074

last edited by wgxli at 2026-10-09 04:48:49 UTC · history