curve #439

y² + xy = x³ − 83284362986612591237124770419410460x + 13323832744501378591680462595599415504584388273763600

a-invariants: [1, 0, 0, -83284362986612591237124770419410460, 13323832744501378591680462595599415504584388273763600]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 4479899643408242486142354247621158610090019978552603894717305621336633370665189642410
discriminant (Δ): -39718810606077930167476934805655304641592918506960890175061918369101870306860126560331166758723567360000000
Faltings height: 19.1562
naive height: 253.5659
primes of bad reduction: 2, 3, 5, 13, 17, 19, 29, 1145843, 23115300343175592012955584317, 46299734573614272920775317733182071773759047
regulator: 11804621917922556586224.516047569157520774637925201678865666370015413517068129017
submitted by: Bhavik Mehta
submitted at: 2026-08-30 04:30:19 UTC
last updated: 2026-08-30 04:30:19 UTC

Witness: 21 independent points log in to add more points →

(329798104056396770, 147403828399081166378066990)
(424928882919214520, -233796912715477087246295260)
(24476695543403690, -106301321488297860080019280)
(-152374516666820680, -149921368610676511470148060)
(43470204711521720, -98922132462493294323594460)
(192955439915201966, -66616368506507756325344398)
(101424898593814460, 76942079631161937961470740)
(-11662518825397480, 119555643644393067208440740)
(2925049295655794120, 4979576424018922912674159140)
(-258382605425594440, -132638737960657090307470300)
(1373414583542027000, -1577836281627999080523318940)
(67490324139288387980/1849, 8081800470536829881179691405180/79507)
(1973813411064263624/25, -10636370328559798070548768892/125)
(-109196171190889690, 145313910542961325479542240)
(279542233094883800, 109026808874265836016048260)
(4238657985224151920, 8707064841694210311608184740)
(205346435383761620, -69861014805627787403608360)
(415172138240621192, 224296315195107401444458532)
(110837536643613080, -73854138477841865377696060)
(118352924598540920, 71587033087488666961824740)
(-27181594296393808, -124769995123970045786416468)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -15/19. The 4 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 >= 21.

last edited by Bhavik Mehta at 2026-08-30 04:30:19 UTC · history