curve #2956

y² + xy = x³ − 915857686161890962086214266778325895920x + 2530612011019132213668450834761707845113339673746570387712

a-invariants: [1, 0, 0, -915857686161890962086214266778325895920, 2530612011019132213668450834761707845113339673746570387712]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 70656293831024198286846918376860534990
discriminant (Δ): 46399369155143865505803369392250252718070351169980327577910749611801313088721940078078097565498300413108305103613952000
Faltings height: 21.4608
naive height: 280.7524
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 53, 59, 83, 139, 163, 1229, 5581, 22109, 330997
regulator: 226383854.265109903978594051221292349289408307146774899
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:02 UTC
last updated: 2026-09-26 20:30:02 UTC

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

(293748129982215501904, 5008036257268139104823012462608)
(93998699855862322877474464/2076481, -694268526526675016423312142430703030192/2992209121)
(28087767020294462941207477504/499477801, -4007086111116240394514291254830058312370608/11162829374549)
(118286889942932031314848852/159201, 1285421960971572207167704616237514171892/63521199)
(4875742252337175768784881001/14684224, 339050860586785415687681283040009245158689/56269946368)
(-193708742141956797841248863678336/8514659804121, 2675017216595180300023551398954629102822844261072/24845683647167232669)
(398694355130542037583836224/9247681, -5843751973411416937146463842092411716432/28122197921)
(88210092206458947251654437365856/91471768249, -828064310165505196244252513530045178053234963664/27664996004532307)

Commentary

2-isogenous image of D-z6old678-093, rational kernel x=-1393367962100335136. 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 20:30:02 UTC · history