curve #6190

y² = x³ + x² − 16653977087916067512824100605x + 827249781988100020096990567888085247623103

a-invariants: [0, 1, 0, -16653977087916067512824100605, 827249781988100020096990567888085247623103]
c-invariants: [799390900219971240615556829056, -714743811637723213709201170482749347287335296]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 34984038809381284469645069641794389460
discriminant (Δ): -15797263861731380822963459960674246516339919161743551926196641403332823986764000000
Faltings height: 14.8694
naive height: 206.5610
primes of bad reduction: 2, 3, 5, 11, 19, 43, 47, 97, 101, 103, 199, 450391069, 803302864981
regulator: 21963761062.8355936252292182521870565345547111607971295955992
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:31:24 UTC
last updated: 2026-10-30 15:31:24 UTC

Witness: 11 independent points log in to add more points →

(-110307304201799, 1149834233014007153250)
(69967143883801, 67348441681155612450)
(74667778773301, 5279080045784841450)
(75077577762373, 9746868410598431178)
(81181000661941, 101364983747156273130)
(94274364316681, 308358194976702999810)
(148328983460701, 1272965255184665715750)
(21455718353173189/289, 30244488478524700464150/4913)
(1567581698447213341/21316, 47222823388837019216314725/3112136)
(2455749072266293861/32041, 186056791715091988178240610/5735339)
(9299787670558257370009/120846049, 49292226331237030940428072973310/1328460616657)

Commentary

Specialization at T = -748/2021 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = 5098973881T^2 - 32101629380T + 51085836400, B = -2550648011199324171649113600T^5 - 231394683896910015292127210400T^4 + 1324630129545679262439545334000T^3 - 2330371426153230648960353010000T^2 + 1239686628515660725984584000000T.

last edited by Andrii Prokhorenkov at 2026-10-30 15:43:30 UTC · history