curve #2979

y² = x³ + x² − 2760743900825445772229469527947200413460x + 55832526931886062186092010081594280532925797601510147899108

a-invariants: [0, 1, 0, -2760743900825445772229469527947200413460, 55832526931886062186092010081594280532925797601510147899108]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 7055360786364598402324518744235380
discriminant (Δ): 71677001542998715805813577566805591015899130827565701836832935579786564910297363713095057078990332128687500000000
Faltings height: 21.1597
naive height: 284.0626
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 37, 61, 79, 109, 127, 1327, 4561, 56737, 354463
regulator: 99740433.3749862864686127940160681230250306794048032857
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:36 UTC
last updated: 2026-09-26 20:31:36 UTC

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

(2033588105189338273329/64, -7085019067904267265805245441225/512)
(40171111099753676952, -98768561514561855151517469330)
(1476256891551104256829864/8649, 1717121694607716802499559293085853750/804357)
(483398189937930780201/16, 75128938537304502834280171875/64)
(5570634871800798200697504/27889, -12732521462278219587042011987338556250/4657463)
(1359634272845696348277539776/22838841, -34937592307889392359571035660636727798750/109146821139)
(3179796397945733135048016/292681, 26076288001959948673694541244376418750/158340421)
(-57048018653767358464, 166331188492054551188794593750)

Commentary

2-isogenous image of D-z6old678-097, rational kernel x=-15169814843694687956. 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 20:31:36 UTC · history