curve #450

y² + xy = x³ − 2010468055035388719404423460545238x + 23314207811821692898503348621971264563108905459492

a-invariants: [1, 0, 0, -2010468055035388719404423460545238, 23314207811821692898503348621971264563108905459492]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 888735090626810706944509868941940721155142515753300241090222766850970045547741350
discriminant (Δ): 285267030962999808448346776054780805713244475992717208279950427261223743384885341566438287380398080000
Faltings height: 18.1713
naive height: 241.6646
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 11189001176854786133142344346792976666120804409872424249855031759053423521
regulator: 661837268961667135129.059658574929550524616910143826827095640318820065520088125
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:02 UTC
last updated: 2026-08-30 15:55:02 UTC

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

(-27130170396407988, 7608514413102741553674714)
(-49690319540740788, -723364156166152663502886)
(12414141352712652, -518775252997634955515046)
(2877267214646916, 4189674678895512712380930)
(3061994932917132, 4145705427743406649817754)
(40695290850014892, -2982175282721220884478726)
(104594448819460206828/2809, -42039800370854474037556715502/148877)
(1106459741569447356, -1162921170261588101078439270)
(38064547469141622, 1392388564597634368757694)
(45164272108636772, -4963817305717431419719006)
(-17391069963921108, 7281379083316975217309274)
(1105529654237728044/25, 571238285332346280530269362/125)
(-7789497035479068, 6205006465404777931800354)
(52237049297374032, -7799552073539349075853446)
(94124781432133452, -25845220892620229601780006)
(41540581832771112, -3388429057648150711235586)
(9519991137177612, -2244408333158575931327526)
(1269835370373966396/25, -903475986468587223207559134/125)
(1999165719604383148/9, -2771539088816321101319796482/27)
(3231549458018590072/81, -1873652719933011007318645954/729)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -7/16. 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 15:55:02 UTC · history