curve #2991

y² + xy = x³ − 7953422367098254760113574496415645x + 267221035087068313304364791938920747056838071167025

a-invariants: [1, 0, 0, -7953422367098254760113574496415645, 267221035087068313304364791938920747056838071167025]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 4455223827985669672160352345650430
discriminant (Δ): 1351120646548266910856737179168576350210738298342613665574246575612937385098494025776197285591616000000
Faltings height: 18.3813
naive height: 245.7903
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 37, 79, 89, 139, 269, 409, 1753, 33119, 1642741
regulator: 53409781.4480988163994516708847476403167415051451924111
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:15 UTC
last updated: 2026-09-26 20:32:15 UTC

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

(1292163218254234790, 1465433947207812873574939655)
(154413713455433126, 52162070278733879693923847)
(4712750430947330, 15160580813180931931243055)
(152566862651188067396/3721, 719466494376242882201469607217/226981)
(41033019185609029216851410/61105489, -260633950869302010123432314511867640985/477661607513)
(312954106606284232490/5041, -1277027468301964716062429204495/357911)
(67760242934248922, -6277977905739570075526009)
(1022063445780343639730/17161, -4920598597114467422556998361995/2248091)

Commentary

2-isogenous image of D-z6old678-111, rational kernel x=-361915575570613521/4. 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 20:32:15 UTC · history