curve #2995

y² + xy = x³ + 2786796855297762902114894534137804x + 240743494205144952993498628495999089260309626577040

a-invariants: [1, 0, 0, 2786796855297762902114894534137804, 240743494205144952993498628495999089260309626577040]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 872082441667970380560815798196270
discriminant (Δ): -26422756900008729269167737768832523043639039850408083788459941994277902784459288073316528468195888640000
Faltings height: 18.5258
naive height: 245.5388
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 59, 67, 97, 139, 283, 6673, 7417, 68071609
regulator: 1288921605.29204913961482347719344906696197417403632380
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:32 UTC
last updated: 2026-09-26 20:32:32 UTC

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

(-44489218886122442, 5357610152250217904461096)
(1184800050303888552/169, -35468135772553303018122939588/2197)
(25649756154191010531076/189225, 4589750560978766838163586206352096/82312875)
(135620725224439269448/3481, -4150769053981794724666084068316/205379)
(2361837048608695735552/18769, 130711126029938156877768376786988/2571353)
(106973264834766184, 41987869729929137582229196)
(17053199443056312784, 70422412414350208849070176420)
(-1184719070202218699403399842/35324826601, -69492948462621391357003910783021032202446/6639265834831349)

Commentary

2-isogenous image of D-z6old678-029, rational kernel x=23812609303061224. 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 20:32:32 UTC · history