curve #456

y² + xy = x³ − 1205908701884023972043061821726860x + 14443705704500032075948025770184419656032855664272

a-invariants: [1, 0, 0, -1205908701884023972043061821726860, 14443705704500032075948025770184419656032855664272]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 146613446594376348938349836157250876838301559148938669816385485863579137947530
discriminant (Δ): 22109580904243323101474980159758459092807147453162165592143262171871673898004061991715263307929600000
Faltings height: 17.9867
naive height: 240.1312
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 61, 617, 6011, 543210295115966222910879889041768196379364320359872798404383979
regulator: 163465550950442862357.987581461528240530403905584604104104506858511252338502823
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:18 UTC
last updated: 2026-08-30 15:55:18 UTC

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

(-19976854285990312, 5528264733209596358404916)
(211925581419431624, 96317071686628716848854148)
(13899094818904312, -606432755174729519889452)
(46943220243993734, -7828259272330146335530192)
(-7877865701744476, 4843014159476813349677048)
(-5238429054619016, -4540597361830297348491692)
(26882112746409392, -1205253781913606174071012)
(-4962145254372016, 4506153296833198914720308)
(28587157664390174, 1825475297199891173214128)
(8635554351354584, 2161941952608761389209908)
(79513143295218104, 20524795628597673805310228)
(45648809845665731/4, 11779135511012382369549079/8)
(285021874077368216/25, 184440730545424867907605924/125)
(4871940471466904, 2946901712725322736870068)
(26271836211361394, -946222783374548104501312)
(118798799412958864, -39342204356988588490211012)
(297935023591385656/9, -88734826172816658752695684/27)
(1196033328913219124/121, 2486364711370641475537091308/1331)
(60392255234864024, -12723217863409701291434572)
(-26695329967193416, 5254679767990263723801908)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -28. 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 15:55:18 UTC · history