curve #3005

y² + xy = x³ − 220312588074858843258305251989281x + 1258652255854945870982607587917870804651796261961

a-invariants: [1, 0, 0, -220312588074858843258305251989281, 1258652255854945870982607587917870804651796261961]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 3075048733534976572545414248070
discriminant (Δ): 4171237431583574120627698643374640684163080048441804917054624376590778471687868766517565849600
Faltings height: 17.1828
naive height: 235.0314
primes of bad reduction: 2, 3, 5, 13, 17, 19, 23, 37, 61, 67, 307, 647, 13709, 29303, 87961
regulator: 9238963.51183484265268656985809241029296499779900671033
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:58 UTC
last updated: 2026-09-26 20:32:58 UTC

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

(165399424187200545665386/19228225, 405093807999023999392023739198379/84315766625)
(6854977819654343069950/434281, 374673062558878804677820948985249/286191179)
(9572394980638078, -163886653036920132550493)
(8556322849230706, 820662434832307957147)
(212794650498411176142166218154/49077772835809, 213310498099684178122088552924025196758753019/343816938723220247377)
(8181826374515434, 61668847653574622544523)
(3098249738425562641792/237169, -90205080975914030240127854568965/115501303)
(2157649605503378554/169, -1594812232056911245075000601/2197)

Commentary

2-isogenous image of D-z6old678-114, rational kernel x=-68458819008845649/4. 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 20:32:58 UTC · history