curve #4078

y² + xy = x³ − 124066041223082908735319590419454919059349362881646x + 531758845149717925116300121723710847941442044136002832593534211537097604740

a-invariants: [1, 0, 0, -124066041223082908735319590419454919059349362881646, 531758845149717925116300121723710847941442044136002832593534211537097604740]
c-invariants: [5955169978707979619295340340133836114848769418319009, -459439642209356287300483314102041140683375355076516957561567731041206457973873]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 428610428388219248912417769724228361243960259613932319147782453962458759457805385527477270
discriminant (Δ): 63459447504799102863292256147010057536178349709496928220268873236565239447334194733857296612776664689614612874855195827692505821220757206591550259200
Faltings height: 27.5305
naive height: 357.6483
primes of bad reduction: 2, 3, 5, 13, 23, 37, 53, 61, 71, 79, 109, 137, 263, 317, 367, 661, 1109, 1237, 2141, 3253, 3881, 8011, 11027, 12781, 15731, 17483, 28001, 39023, 894227, 22707917
regulator: 229676653253281008959036.074647951944341475534026264752044499123880521083225094
submitted by: wgxli
submitted at: 2026-10-09 05:29:49 UTC
last updated: 2026-10-09 05:29:49 UTC

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

(6344139792727965011797196, 78552475170257111104054721191398194)
(6230581666474965641621690, 792051188319558805663756309080385760)
(6244335090646645546693244, 725451822056931498122457476040934850)
(6259211570349900133086584, 651875843220948707642289744722631170)
(6297158699354622189394748, 451811084420579506444050747479818946)
(6518471416673440619624048, 103986974977731664991464292706428846)
(6544756981164401056053308, 337423920755368521337676555483721986)
(6612714384043993557422468, 711525350762585933801314166727799586)
(422555434538898459692986, 21895422210359935092846962391077298790)
(4932629692385669178004604, 6308883334042835408933667199655907266)
(5512987884717678165743996, 3916720441378800967834537425068409794)
(5601539081965749843286618, 3543854275248946762823062190411964646)
(5954027814656164850298248, 2034486084831621543503723739865066946)
(6101126983894758557500708, 1386723373589166790450912359193544506)
(6113155516196273438531708, 1332876798945381100368861830411726786)
(2066443557154303439599868, 16858454202495138677817148911340401986)
(8850199040788374470401148, 11267234811211080295899306393393246146)
(-1016779353531021070992910, 25629191304347152918413345204837910280)
(-115751108734402879081906628/9, 2286195628876847653995861911382011222/27)
(3223813941509603561015114507/484, 10069386212699873495994022718662247130233/10648)

Commentary

specialization at T = 2783/971425 of the rank-9 fibration of X808 given by: y^2 + 84(35397T^2 - 221T + 20)xy = x(x + (1139T + 115)(1269T - 59)(3776T + 34)(539T - 5))(x + 8(638T - 23)(250T - 1)(7811T + 499)).

last edited by wgxli at 2026-10-09 05:29:49 UTC · history