curve #2898

y² + xy + y = x³ + x² − 273428596724487092047800235x − 1260865140524751073930533279856028245735

a-invariants: [1, 1, 1, -273428596724487092047800235, -1260865140524751073930533279856028245735]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 23959330209511704615223770
discriminant (Δ): 621528005658546166397145444428078137593428577802111230401472503454441061776614400
Faltings height: 14.2104
naive height: 194.2329
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 31, 41, 43, 97, 193, 241, 2087, 2713
regulator: 3537273.104264385143936780341026460054517525105461388
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:27:25 UTC
last updated: 2026-09-26 20:27:25 UTC

Witness: 7 independent points log in to add more points →

(-5450206960797, -8214632578091546042)
(-8042370591997, -20444326455595091562)
(-19467497201816965/2209, -2237275045696324381194438/103823)
(-302099846091853/49, -4734839221643410091766/343)
(39522109920803, -222858907300697422842)
(366159024260575/9, -6329540314689979670582/27)
(1045312272628687/9, -33438288140499352950194/27)

Commentary

2-isogenous image of D-z2z4old678-054, rational kernel x=6694286189203. Source: https://dujella.github.io/tors/z2z4old678.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 20:27:26 UTC · history