curve #4063

y² + xy + y = x³ − x² − 36110387659212751905912184800098827074197970280996697x + 2281891795150649327386169740769122327971517159511941226970041612636804891194169

a-invariants: [1, -1, 1, -36110387659212751905912184800098827074197970280996697, 2281891795150649327386169740769122327971517159511941226970041612636804891194169]
c-invariants: [1733298607642212091483784870404743699561502573487841441, -1971554511010161018861650648224677956977436414141285303280769305291437845296475761]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 3837790555226969705537890168396950553913036309626451197868514544063254214128052127345730
discriminant (Δ): 764099262650684849672633829624708925630842407000252083204662584628011249477802993841592848621305996923811493921258105978769784891816631847192444861929881600000
Faltings height: 29.2117
naive height: 374.6689
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 47, 79, 157, 197, 239, 283, 331, 479, 1381, 1511, 1531, 1811, 4111, 8783, 16943, 20107, 32653, 36901, 135409, 17470571, 220570313
regulator: 12746766710868894028145.8426513966906738917935723267104903472851657574773564092
submitted by: wgxli
submitted at: 2026-10-08 22:48:03 UTC
last updated: 2026-10-08 22:48:03 UTC

Witness: 20 independent points log in to add more points →

(74761088215882442469470503, 9803601168587528224069543093705673388)
(74761908203915691129124777, 8958233209269883027798539821715513286)
(74471054396781312964779927, 75663714715314606043009805241037293436)
(141630143524136830589989287, 92440650670301906282230240106703928876)
(42746352782766638527494177, 903555602703044653541406892373896539886)
(51909761285723148437934327, 739788784013998398858543191240343086236)
(52114929551846244384269187, 735896634600698034599577237565217561836)
(66819099337371668547396327, 409098518244791269037965449229073619436)
(68556018206266656188321767, 358490900412051730877504861438419048236)
(70289292655657169976948999, 301642819013902440410420658168547019692)
(72100996912140139915055527, 230472892823935346797320643358014779436)
(73469951896633085388553527, 159505306557649240649224573160413747036)
(-2724383059560738872997053, 1542805918602807495125878604135984595436)
(145071022257426966091155879, 310528367097348217902277072722314737708)
(147484236401516359735573479, 405212240256504121123842060215435672812)
(148103003328569733159707287, 427083361865705244821618596022079731836)
(149331454879852444096263207, 468560672663528084870974283995920440236)
(18783227520944600325938167, 1268955898187418215071825469314267229596)
(19592694610079009358035847, 1257741275984062729509614577388630400716)
(162163751185882459144920162763/9, 65299011007905183468793630021953775696571872/27)

Commentary

specialization at T = -9229/194873 of the rank-9 fibration of X588 given by: y^2 + (1744T^2 - 8532T - 607)xy = x(x + (63T - 109)(647T - 41)(7T - 2)(T + 95))(x + 9(111T - 40)(383T - 38)(87T - 16)(T + 5)).

last edited by wgxli at 2026-10-08 22:48:03 UTC · history