curve #2962

y² + xy + y = x³ − 89800461052237943201152027335174416783x + 327180041942345393885215345805726536326256241541230408306

a-invariants: [1, 0, 1, -89800461052237943201152027335174416783, 327180041942345393885215345805726536326256241541230408306]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 3812079870065732122018767404589934530
discriminant (Δ): 102155643642707638584323698967601795759877507474928171604550839943691225762114094199006106602570080227649278250000
Faltings height: 20.5925
naive height: 273.7856
primes of bad reduction: 2, 3, 5, 17, 29, 31, 47, 67, 79, 199, 269, 617, 859, 2539, 5407, 85808363
regulator: 497146176.446804164467731024618586934915942338391945133
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:32 UTC
last updated: 2026-09-26 20:30:32 UTC

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

(9654599117446146915, 18976569002992092038027175892)
(5879280510277370115, -1562304332719466085943530008)
(-10940884010435854149, 143024813315446773570559924)
(12634472754741182212995/2401, 69254096845812086166278866990408/117649)
(61105948739939193607707/11881, 1515350804145806700298991791813096/1295029)
(3350327435872072973596495500/373223761, -112726671639541449979769632143354121537897/7210309838759)
(1055471153248644673858870345755/187443104809, -20158777152426029225501083455844479744975784/81152929897742123)
(678008462173179290, 16328081542322324711249989017)

Commentary

2-isogenous image of D-z6old678-032, rational kernel x=-44950628272970786809/4. 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 20:30:32 UTC · history