curve #4061

y² + xy = x³ − 638703932228882840978291758076199757002097637073665x − 2000682293681461752792191697520052233697713603839721993736989193429426395383

a-invariants: [1, 0, 0, -638703932228882840978291758076199757002097637073665, -2000682293681461752792191697520052233697713603839721993736989193429426395383]
c-invariants: [30657788746986376366958004387657588336100686579535921, 1728589501740782954412453580670642009435260003280513221102376158971994536307031]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 2068433157780159063084093615556099330453046637971536818742205532992224375804129639558855110
discriminant (Δ): 14946316151938412093963886923582110367207761154009221689493780190885752775897916361128842174340193262341561106853097569140330157857920390191514410000000000
Faltings height: 28.2753
naive height: 362.5642
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 31, 53, 71, 79, 89, 103, 107, 113, 127, 151, 271, 283, 293, 739, 743, 977, 1471, 3191, 3617, 744701, 1489951, 14742361, 28928843, 1705548529
regulator: 7937150229384211320267.90339697123368144714642051803445594914616186298132370768
submitted by: wgxli
submitted at: 2026-10-08 21:44:26 UTC
last updated: 2026-10-08 21:44:26 UTC

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

(-23530128896536187096562326, 498482997877693975054565702639289163)
(26724161812456242995140774, 4046528211539539320737169270911091463)
(26757874138753429018027174, 8195330511934246477812054349616430163)
(-5988230243850446346844076, 40115988732221052534832539060132735163)
(-6168081239843223783846842, 41282313436135790644735110899166218131)
(-6710533740660202967530016, 44532888784804402255869444891539456323)
(-8487257481536782006770686, 52998068229711501107416512006873373603)
(-8653314911481750925525526, 53649463809180872801975325946595783563)
(-9021957778121033914612676, 55021167263600750624831527752911087563)
(-9780587713623833469429326, 57537879130033592163953798654050278163)
(27585363237176219939111674, 37034880436325929736962597413234081163)
(29337421987856213916950674, 67168819226629226591167043798086818163)
(-20451159919006360924468166, 50078513502332318145719276597884333243)
(-22310842767756921207636326, 33817005294351990379271133298122297163)
(33287491773040592069301274, 116717212532811949090114327427143410763)
(47281118205819088413716374, 271104808190343481298434892304974999863)
(53901757114096860902182924, 346667257379554282137483579560737197163)
(80132503007178679446708274, 679239756412775944639239900865592443963)
(82523652386913173356701274, 712242832237452131491387267226785237963)
(281374408518341113950040954, 4700546203931712840415696986666928324603)

Commentary

specialization at T = 88516/98303 of the rank-9 fibration of X552 given by: y^2 + 238(19T - 6)(5T - 3)xy = x(x + (12T - 29)(51T + 22)(79T + 22)(482T + 29))(x + 256(T - 22)(83T - 7)(21T - 2)(7T + 17)).

last edited by wgxli at 2026-10-08 21:44:26 UTC · history