curve #2952

y² + xy = x³ − 2191812816396155618802462282685422663318x + 40434428546253979747278064136169405000957468123519807816228

a-invariants: [1, 0, 0, -2191812816396155618802462282685422663318, 40434428546253979747278064136169405000957468123519807816228]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 31206133418171078552200701249480523698
discriminant (Δ): -32403288764572216931414588263712638049740601000366815099391473313771046682830609674059662848714421123093613254722360832
Faltings height: 21.5207
naive height: 283.4172
primes of bad reduction: 2, 3, 7, 13, 17, 19, 31, 37, 41, 43, 59, 89, 97, 103, 241, 2179, 4363, 9349, 77867
regulator: 358821411.898662606422013888483693292567617787643322601
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:50 UTC
last updated: 2026-09-26 20:29:50 UTC

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

(197547378825372012240543252/5669161, 1077328151117331710958814845957071546178/13498272341)
(12187465315262662023217713294/45495025, 1326187970352916428834669361813308455781582/306863943625)
(219173891807937428377075961262/7745584081, 22304646157851209166676353575010107950355892/681681109384729)
(848516061670468129121922085302/6196680961, 741027408334122281276776159366530147363663540/487796528568959)
(32616167912949789870541294086/1659666121, -4757146412030640116956789443092627752189302/67613138103419)
(-29685007250922343695825536064/828000625, -6434413391983354586445226176394330550728266/23825717984375)
(19829617797827890931288132951476479673639394664143078861480214/37839954002796389975258106067756400087041, -87961561305317918747478189995530840375920498408288545463619444319515239599358447165836681580/7360814949365290411721864974032032103217419155525365598627839)
(-133997709194063451913582276120558387290583120056945264738/2578561569062255695772961349780080481, -489950108797155448661918604335260782437306644184991177395655515639621134021016314692/4140628418084450332678003652314874689518425305375573871)

Commentary

2-isogenous image of D-z6old678-117, rational kernel x=27100800076051676620. 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 20:29:50 UTC · history