curve #2988

y² + xy + y = x³ − 14520399801311951709616847578451908x + 669388104557948345430083778428419593473257091288306

a-invariants: [1, 0, 1, -14520399801311951709616847578451908, 669388104557948345430083778428419593473257091288306]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 10893096871712594532803840282233410
discriminant (Δ): 2365910575324213424813059887140281073007254916373578823326758516049737704712532211172619024030842250000
Faltings height: 18.4768
naive height: 247.5962
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 53, 79, 103, 109, 139, 541, 881, 395449, 6383317
regulator: 871286209.097747641201563153731438860528025668943951093
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:04 UTC
last updated: 2026-09-26 20:32:04 UTC

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

(1092183078002612626, 1134740583352845938272913594)
(97398751043592226, 13382607248208989778944474)
(159612118109445565, 49173561577405003708766267)
(-86371900913352463394210/1203409, 48358073772299122493929884053823106/1320139673)
(212505406183281638330965/17131321, 1571469583620263628472679862078269473/70906537619)
(-548637234599776003966010/25512601, 4016962754908996964789274463405241542/128864147651)
(91104670638338325961165/48841, -27442606796538030984702731472105713/10793861)
(-30739813889800139880833710064540/783918074140369, 753472249017791376142614887102965025779801276726/21948559203787864640153)

Commentary

2-isogenous image of D-z6old678-078, rational kernel x=-591567337254347721/4. 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 20:32:04 UTC · history