curve #2989

y² + xy = x³ − 290635083391475271382480377860066906x + 109086645887415145506770760073398249692342985043954020

a-invariants: [1, 0, 0, -290635083391475271382480377860066906, 109086645887415145506770760073398249692342985043954020]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 3808846288107559247843042095265490
discriminant (Δ): -3569581913939795835997155844150593780538971022332362278017744221776643606121783131922966479177111138535360000
Faltings height: 19.5222
naive height: 257.7711
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 41, 47, 61, 73, 83, 229, 5297, 8311, 12589, 25121
regulator: 9158024916.44782972458412998455555012122390490942159593
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:11 UTC
last updated: 2026-09-26 20:32:11 UTC

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

(-3767907115522726193204/16129, -830121082603601301352572112092350/2048383)
(449819005653722697466927728/1211805721, 9651790493163744918806665403538798096846/42184168953731)
(-19706032594169077904/49, -137596140756980055663109728230/343)
(25332499612387068216964/63001, 3783134988568558970592666839756066/15813251)
(-661673219306550805397076/1901641, 1075127869341904164209137110119175474/2622362939)
(-14449026451492185845616613499936/61497544309369, -195542146831001422715289339739673609568088161950/482266017883211144653)
(-32110667471192912733372/212521, -37887566255166288203004026916913554/97972181)
(2700985152934516456547458084/11773335025, 298186146754945266551759530389181777218838/1277465716887625)

Commentary

2-isogenous image of D-z6old678-023, rational kernel x=336435780064188244. 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 20:32:11 UTC · history