curve #431

y² = x³ + x² − 42135622063493348388166878576380x + 105877466660516913227884540253347402151079039600

a-invariants: [0, 1, 0, -42135622063493348388166878576380, 105877466660516913227884540253347402151079039600]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 2265959254716445430438863265989168698042215532450755065489869558062322360
discriminant (Δ): -55022343847835744641882964746299383827848669540032026520029197872137136857827875016113503200000
Faltings height: 17.0145
naive height: 230.0804
primes of bad reduction: 2, 3, 5, 11, 13, 17, 37, 71, 109, 103577262275125889, 261899634925063603879508896992741770234315969
regulator: 28294149260330363850.5939340333993450633051248226970528120206470659169584483158
submitted by: Bhavik Mehta
submitted at: 2026-08-30 01:39:37 UTC
last updated: 2026-08-30 01:39:37 UTC

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

(14876401308396490, 1664724447800859994713450)
(-6725532642415010, -291628817419821723076950)
(1704793802153590, -197483161034991856360050)
(124097154691060, -317254577524680272871420)
(-5908640639946380, -385433588839789717664580)
(31920350315417752/9, 874590596283888376979372/27)
(1891403249940730, -181516753954862840908890)
(3363354413861590, 46981013649061522194450)
(-5446028865838985, -416922361789685147960850)
(3290150727445015, -53489277131182399499850)
(11779407378473428, 1115342934804045558034164)
(3666034693317430, -26031239863359457279410)
(7004910280329850, -392993043392181978314790)
(-5816811217160510, 392630762165060182356450)
(2560678840579150, -121541091389674892445510)
(3544125324880600, -32568036387271480763460)
(-3815458612800830, -459455975828420059430370)
(1059524529032446150/49, 1046269943811131894456285970/343)
(-6046497608743730, -373616977585042226019270)
(13629890936117890, -1436541644452482083008350)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -11/6. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 01:39:37 UTC · history