curve #3649

y² + xy = x³ − 409400407501145672374429749475x + 91075658316227433599720258143475077076650625

a-invariants: [1, 0, 0, -409400407501145672374429749475, 91075658316227433599720258143475077076650625]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 177599378108948787912335962451915637757725844372530
discriminant (Δ): 808277286802827723504942151135272525682244753957703943066833547380231096051981542400000000
Faltings height: 15.9861
naive height: 216.1671
primes of bad reduction: 2, 3, 5, 11, 53, 59, 71, 73, 83, 107, 127, 463, 1069, 1231, 2879, 5547599, 16379591, 16791659
regulator: 63230625934718757602.532952998583933132540709118192698487166255661398
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:46 UTC
last updated: 2026-09-29 19:26:46 UTC

Witness: 15 independent points log in to add more points →

(-2335864005896935129745586309284290/4577138393556897961, 126582341530562225240314450610136186980650676336125/9792443305242158231251771541)
(-3715373275101014957436788050/7988983537441, 303697636377659203148802537494153593087975/22580694199922700239)
(-362050639803131808612994/9238093225, 9187296566565440473676110067365837859/887919330320875)
(-11793695147606852029090/152448409, 20814707151874450193847993771795355/1882280505923)
(-4283243745082981816450/15031129, 791774944740445679217558893884325/58275687133)
(-9701360227697093895119/89113600, 9750853930743964047723760556198759/841232384000)
(2798858178321520750/2209, 4161187146005479234094289575/103823)
(25004065108141168942/159201, 351666882944680178622652662919/63521199)
(43419043896221025902/160801, 30055346315835942653521815577/64481201)
(785852881184857361684/1481089, 8687004856134712864262184010531/1802485313)
(3837122899092591007550/16008001, 165963834110824936130665071978025/64048012001)
(143335895999980615957310/559369801, 22892321981730785223509142387907315/13229655163451)
(430057826317780571974400/362635849, 246432198431566791348182068112357125/6905674472507)
(119465119438878449676903550/28778747449, 1290991664718071927770975679373182033675/4882113053490707)
(568388326843547054465577609950/2344208697186841, 279512331533406381398373490330897139737214725/113499620473747499115389)

Commentary

Specialization at T = 471/107 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:26:46 UTC · history