curve #275

y² + xy + y = x³ − 2034488389107661074627844285x + 35847670110541831966937994064437784692732

a-invariants: [1, 0, 1, -2034488389107661074627844285, 35847670110541831966937994064437784692732]
rank (lower bound): ≥ 20
conductor (N): 42943483208607336815574462222443765285847682460232958112909965535488718
naive height: 200.2833 ★ record for rank ≥ 20
Faltings height: 14.5722 ★ record for rank ≥ 20
discriminant (Δ): -16197498753307130570718466689601294928007257164818990602909253701371655741958401236 ★ record for rank ≥ 20
primes of bad reduction: 2, 11, 13, 41, 33882942493, 2222460391457, 19721425437778783769, 2466004856513281132232797
regulator: 102030176007090039083.688036283641391460542002868509037467276891445799630022015
submitted by: David Renshaw
submitted at: 2026-08-20 16:56:01 UTC
last updated: 2026-08-20 16:56:01 UTC

Witness: 20 independent points

(-286935581018, -190870148372628414290)
(52969417814370, 276950216682633411746)
(7583485767200, 144413558575024721811)
(27946486312659, 28587472751628447056)
(38324958975646, -119028506800841604138)
(5466090182467, -157766569909090409868)
(953813852774, 184141289418161265579)
(403282323072, 187155717037235338220)
(-9329617701204, 232414725785122010816)
(20514524074244, 52388521659913741416)
(-667941066879, 192889322114541114462)
(37221137375463/4, 1064989158999437585781/8)
(-469358923866320/9, 285554907899126306516/27)
(32416674872899, 62937145652566229704)
(-42650906952935, -212212575043297396194)
(-663480184290155/16, -14153580123795004605281/64)
(228058396114600/9, 641761265157468659201/27)
(517091678959083/4, -11123125931505983025005/8)
(586711235002, 186156425539390387690)
(-31497237573875, 262070200192434671788)

Commentary

Generalized Mestre/Fermigier quartic family from Mestre parameters u=-3, v=-1/2 (canonical primitive roots [0,113,550,753,868,1058]), specialized at T=10239/176. A native M=6000 Nagao rescore selected the curve; quartic rational-point search to bound 2,000,000 plus an integral-x search to 10^12 found and certified 20 independent points.

last edited by David Renshaw at 2026-08-20 16:56:01 UTC · history