curve #5954

y² + xy + y = x³ − x² − 3524981191860076702011843898859541587x + 2547464059667601974959163519855253999787085204483693299

a-invariants: [1, -1, 1, -3524981191860076702011843898859541587, 2547464059667601974959163519855253999787085204483693299]
c-invariants: [169199097209283681696568507145257996161, -2201008947552808105603321343713162888181483334520250027841]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 9540524621154266900569830906607102382329164090
discriminant (Δ): -319465164959143229842920421177179570129081779373334332207791466547301885006700661068529506025734144000000000
Faltings height: 19.6844
naive height: 264.0725
primes of bad reduction: 2, 3, 5, 7, 37, 149, 223, 283, 293, 613, 27487, 177787, 612583, 2998252970539
regulator: 28251256440676416.042089829499206966906511684895427355824009085
submitted by: Maksym Voznyy
submitted at: 2026-10-26 10:55:07 UTC
last updated: 2026-10-26 10:55:07 UTC

Witness: 12 independent points log in to add more points →

(12760590949039390694604465763/9692993209, -414334095338912830552064168001078134689958/954304260405677)
(-55904098852840974838830931573/87923517361, -55499330796103270720157750228502973608491986/26070993444366359)
(5067200076952064774708027/5612161, 4221770941686571581635146566923493314/13295209409)
(7667348508113360020434067/7070281, 227071526570733200075855483009391734/18799877179)
(115915383412896405852314132867018827/88518965011634161, 11080514879533349282741156651748315029619815760839314/26336283652724496507886409)
(-76855956265554349611869/822649, -1265359785006847766230662342667945114/746142643)
(-284202805762907722024944877757/137572002649, 51358664329805495256922511562168019420343878/51026418786532643)
(-26177651889790186715665037117/13059232729, 1863308008277591969977229986738020819861818/1492369938571933)
(3472412303699687139724376130883/318306970969, 6380526016229326739013097614693684168240160838/179584655030087203)
(803595003037040070543296491/625250025, 5850420540440019617332503373085861040146/15634376875125)
(923331021523599387366592588363/732912346609, 204407569801181805380319120511393039207768942/627448458669004727)
(1005446022721576165984586214476083/858438256979689, -4022810264951221049817661458191407675332087187222/25151482928523974634613)

Commentary

Voznyy (2026), T = -10375/3108 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-10-26 10:56:14 UTC · history