curve #2941

y² + xy + y = x³ − 449925682880949330498987645240846438047949x + 116160476661600320820319878143004063014637083173424102420727556

a-invariants: [1, 0, 1, -449925682880949330498987645240846438047949, 116160476661600320820319878143004063014637083173424102420727556]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1677463821752304934698337236145168971930
discriminant (Δ): 24289451110987625023862427826771956438895085379792514010614818751375454605447515713681370106273981336814703886328528236980
Faltings height: 22.5324
naive height: 299.3433
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 37, 47, 61, 67, 71, 139, 409, 461, 467, 617, 1297, 298023589
regulator: 10000596066.9425366472539586613928691009957529608702088
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:20 UTC
last updated: 2026-09-26 20:29:20 UTC

Witness: 8 independent points log in to add more points →

(318413273180740146632591/676, 51955146227966948577486758991353757/17576)
(1150629815422306532729, -33493846191407521736385841355508)
(17951377187512237894834210333961/46238761024, -288640412264517167239711984751737848230348079/9942813260512768)
(408215406416605293342089396/1050625, 43881676831307244782315405505136328619/1076890625)
(5733727140257291537636146709/11262736, 164970788241859877865861067226086622341617/37797742016)
(104408587905502417781978081465381825/268850065635904, 148171354285328663134170970200685902851069583625749/4408239891008814049792)
(354337415641150212071, 1106258848750662910638941237592)
(-1500887335769872430957644800237955901/3282689294278564, 2829319793995243338113154755320366358730749457947162509/188080888704881108967512)

Commentary

2-isogenous image of D-z6old678-036, rational kernel x=-193404925306860615997. Source: https://dujella.github.io/tors/z6old678.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 20:29:20 UTC · history