curve #3066

y² + xy + y = x³ − x² − 187182872354048710166464127x + 789278045666808478472233792388257219751

a-invariants: [1, -1, 1, -187182872354048710166464127, 789278045666808478472233792388257219751]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 110072036541366600851273211270
discriminant (Δ): 150619358850115477130021527951035002334995235897056476256679076854947226683040000
Faltings height: 14.0998
naive height: 193.0960
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 37, 41, 59, 67, 257, 3203, 48539, 173659
regulator: 83182465.102554224727810228893765146290598724799199290308
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:39 UTC
last updated: 2026-09-26 20:35:39 UTC

Witness: 9 independent points log in to add more points →

(447638361211219/9, -9137078726996872948622/27)
(3720797550501, 12013342315304651074)
(12526540512981, -20251339821794173886)
(48808621503591, -328525545087433361006)
(385014297035422851/7921, -230094619899247531126008874/704969)
(-33549787569, 28205636926931620714)
(16494172313806395402081/384905161, -2018654839811212368008972399987194/7551454353659)
(364543020725225611/34225, -12566897855809576170206914/6331625)
(694734213184393646151/62710561, -4308125942303475270151884519154/496604932559)

Commentary

2-isogenous image of D-z4old9101112-042, rational kernel x=7718659159251. 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 20:35:40 UTC · history