curve #3122

y² + xy + y = x³ − x² − 253117693608320317481471859070333603713560604347x + 48928612825281236145946031275063726668034883245591529738101538876498219

a-invariants: [1, -1, 1, -253117693608320317481471859070333603713560604347, 48928612825281236145946031275063726668034883245591529738101538876498219]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 66720045390731263315331475333869520786441558783362426910
discriminant (Δ): 3668869303550805917697384026448682337766848072601000322829128847051741099975481846658783305595011420200893411393993319896752921344768778240000
Faltings height: 26.0500
naive height: 339.0642
primes of bad reduction: 2, 3, 5, 7, 13, 23, 37, 47, 53, 59, 61, 67, 83, 101, 103, 151, 191, 283, 317, 443, 523, 613, 1063, 1621, 1747, 2383673
regulator: 753887898795.193143908201459638465286775322744444883760771347
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:20 UTC
last updated: 2026-09-26 20:38:20 UTC

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

(418280425983058874460427, 127421184276024187226510211355140186)
(8276856246345306536968764361339/1274641, 23742677510397535405074378951136151918343766218/1439069689)
(2827023297687111812398003/9, 549146291730148394020487297823463822/27)
(300520540784526948650187, -1528620058749285161768233412982374)
(4542160993231767616966930932123739/13807425025, 57629409225881756136763550761581807792329213639178/1622441477562625)
(2323088150052179069288934981633/7661824, -159617382757634763863008260351496450117768903/21207928832)
(112485099459600231592247181995436647431483/175100086988842129, -28512938986560886458786923043069193476770722647999152097904138/73270565387748194086567367)
(699358954749915120973123269846147/1826622121, -7044379110903680222102627953030141068752927479586/78068002829419)
(3026759980001961350605545853313778179019883/9345848566798351969, -867622611422269842892126128292232478534209178753328112376796218/28571175550650567916022136047)
(-1488870890822375752779100836644373/2861073121, 30501602744268723872071899531822638596109021973434/153035940169169)
(623651649581811709046563/9, 4808854753798866730882142773733126462/27)

Commentary

2-isogenous image of D-z2z2old1011121314-048, rational kernel x=-2403108265506338506958421/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:38:20 UTC · history