curve #6399

y² + xy = x³ − 1597013756923729426505710x + 763188880873084102125914053590356100

a-invariants: [1, 0, 0, -1597013756923729426505710, 763188880873084102125914053590356100]
c-invariants: [76656660332339012472274081, -659395193074459649227288250820776081521]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 2446902750293324912726293074310
discriminant (Δ): 9057399872353381332958931741877083790158814474157544343339008000000000000
Faltings height: 12.7904
naive height: 178.8041
primes of bad reduction: 2, 5, 7, 31, 37, 59, 67, 103, 109, 131, 12703, 144031, 2865043
regulator: 27190845073.94159123776589810055680012481921943198967970367
submitted by: Maksym Voznyy
submitted at: 2026-11-02 01:20:13 UTC
last updated: 2026-11-02 01:20:13 UTC

Witness: 10 independent points log in to add more points →

(109433758350005503420/126315121, 245353778345585029038016315310/1419655644919)
(152541298072017486002186498780/132333499831298089, 32433944112183109162907392631621390347163370/48139846117749672664793813)
(1307367932062814708505129327933633479420/1538171401640850411644160961, -8493701214408342333295361547514670877264629149288098574590/60326356847619743874910962209816946530209)
(183422831498923064/223729, -7437902041942197227707786/105823817)
(2908852342320847580783740/3444858497089, -827234930274933640283269315412983550/6393771050927587937)
(555396443081660/529, -6007305334647098481970/12167)
(2300066159233373180/1216609, 2847008589304062024117091230/1341919727)
(177582718702416380/214369, -9262053261541434999343970/99252847)
(745311037753469883307580/417480100129, 511948165139342187883393361664664670/269745164656050383)
(24450575358956814220/5612161, -116285624441734153180584334590/13295209409)

Commentary

Voznyy (2026), T = -431/80 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-11-02 01:20:13 UTC · history