curve #4112

y² + xy = x³ − 625022887584075587876471652551908168193702224564445x + 5275345238936913307330519537386950572289961377764009412477178576690846479025

a-invariants: [1, 0, 0, -625022887584075587876471652551908168193702224564445, 5275345238936913307330519537386950572289961377764009412477178576690846479025]
c-invariants: [30001098604035628218070639322491592073297706779093361, -4557898286441493097533568925303973200511968957494063116117670400207451526517641]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 13147875829450866457593374848357371457853900269857543652894501277726603180941592610
discriminant (Δ): 3604473119198493926787991306265641595749850564692438365371081852502252882901478858200322536611591459913787197056528663535414570809761417634940570828800000
Faltings height: 28.1925
naive height: 362.4992
primes of bad reduction: 2, 3, 5, 7, 19, 23, 29, 41, 53, 59, 71, 127, 131, 149, 269, 503, 521, 631, 1181, 1283, 2017, 2153, 2741, 6079, 9829, 93601, 137399, 1553129, 3220558637
regulator: 1382948841253494384105.39365813609858421001979882452694301456263444625315011299
submitted by: wgxli
submitted at: 2026-10-09 15:27:29 UTC
last updated: 2026-10-09 15:27:29 UTC

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

(9788772741911599936895650, 9751878833139334298093956062694187695)
(9790577767404483393707590, 9720592946535546789166021762749829555)
(9912864835229380637014090, 7325598985146304189499536296004897855)
(9950564081316083090145442, 6423189598778524460486803201293146287)
(-455678849895312190282590, 74565812078247035805461676008197237935)
(10044857045646575497090210, 3255341992359094246708832552082132655)
(18532545006989751221579532, 7561735854550862383049239696964192037)
(5245982614666791740455240, 46269398401994682895220533462967205655)
(7079834123551839449394850, 34715368925518631029963043681699320495)
(9517638564969211191862690, 13739098002490515042458418468006113455)
(9570784412036466131435590, 13041049069789709148826508445834181555)
(9684477403831351907281090, 11429070476838960088924311394609791055)
(9740440996800203746982362, 10558533012748631799929364070011905887)
(18809272157353060533626710, 13176795716202163350481455198527188135)
(20774599501621262714441650, 35450380792739512234767415665323501695)
(-26956858220679826789570910, 50350759652942781951104131211515323055)
(60887001173757593123445670, 439251073177076208741150747677444902855)
(114695320265133356233148002, 1201002487287482475271024133361643974511)
(146690633293619861655980335/4, 1423787872573204369382160644015553685865/8)
(2014523024433217733192568130/81, 52134188649343685062141403737194267255895/729)

Commentary

specialization at T = 9620/8279 of the rank-9 fibration of X1152 given by: y^2 + (71511T^2 - 43336T - 967295)xy = x(x - 256(155769T - 2147777)(2T - 51)(11T - 3)(T + 24))(x + (200T - 3009)(183T + 17)(15T + 274)(7535T + 11673)).

last edited by wgxli at 2026-10-09 15:27:29 UTC · history