curve #2727

y² + xy = x³ + 975035276216276408308394872382042301448472110x + 301778377978256998309565817006700452968439569814815914186322136680196

a-invariants: [1, 0, 0, 975035276216276408308394872382042301448472110, 301778377978256998309565817006700452968439569814815914186322136680196]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 10350901625769093734552060155302
discriminant (Δ): -39401647266209058723815804751982997086428481283857346578128763306248533191747309216486109598949790555384028064778862611485089621683294911488
Faltings height: 25.4650
naive height: 328.8838
primes of bad reduction: 2, 3, 7, 11, 13, 17, 29, 37, 71, 79, 127, 149, 1069, 7001, 9439, 12601
regulator: 5571803.737943839901793189790143413037260136031
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:36 UTC
last updated: 2026-09-26 20:19:36 UTC

Witness: 4 independent points log in to add more points →

(-1356967137581326215973390036919974281048/202973418074158369, 1570511307579493912116583053433358193395241474767765539278618/91444743897907286068642097)
(3760479806674991392656428855791540/61678219201, -371395243864773365604396200226153122786882856956206/15317847416787551)
(-7702589112135690405202388/289, -78739667894426492600037532945467034378/4913)
(223571536129444351984649172976220679629550252628675932166821648768664449073224246727165641886718548/45851530559281560644273206898280492031378618866337173997233492814961041, 3342910629813336652090975855082954324628693287375318517253743473940210155082815919575638489580531051124883238125879107914691166964191690675757594398/9818174712017894657037452554544713458984231284184349042280184863698837024659905486361263144117661868703161)

Commentary

2-isogenous image of D-z10-020, rational kernel x=497910965560885818914783/4. Source: https://dujella.github.io/tors/z10.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 20:19:36 UTC · history