curve #3026

y² + xy = x³ − 167491337907553149733596148776x + 24129662094279893327932223367787466012077056

a-invariants: [1, 0, 0, -167491337907553149733596148776, 24129662094279893327932223367787466012077056]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 29700487374913942444634217870
discriminant (Δ): 49188405370646308675681916615228543581270463621534619250434652753148207090437046640974400
Faltings height: 15.7564
naive height: 213.4858
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 53, 83, 139, 163, 283, 401, 859, 95917
regulator: 2590324.16777865123719040906312665383245137207124019552
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:51 UTC
last updated: 2026-09-26 20:33:51 UTC

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

(87242841500652010/81, 24103609513125664253483474/729)
(154233749714574, 1402057369900464232548)
(70835303689912186806/235225, -114091836077787524518737156834/114084125)
(-276816607335726, 7020135441693797788308)
(-2740727554175139/16, 442452489937165905446163/64)
(4359465102120925496908854/96963409, -9101896095553260317349749627331827094/954798688423)
(1874871015967568116234454682/14825480049769, 127269345569901656457037877692854876239286/57083835652389910603)
(34274726570413695559426554/3073482721, -200526664807247253590388018108563835348/170390808569519)

Commentary

2-isogenous image of D-z6old678-050, rational kernel x=-88697342525360. 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 20:33:51 UTC · history