curve #6359

y² + xy = x³ − x² − 204864170251808924784x + 1501421194324709686447667932324

a-invariants: [1, -1, 0, -204864170251808924784, 1501421194324709686447667932324]
c-invariants: [9833480172086828389641, -1297227911852298508316394365774565]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 2543688954831820210243429453963981603296447005494401711726
discriminant (Δ): -423569997380685037769315385815174288617717762460916784233190068
Faltings height: 10.7239
naive height: 152.4911
primes of bad reduction: 2, 3, 11, 29, 76675843, 417850931581, 4608924852593837653115938282425197
regulator: 10844315515721242460799254308.75717777865826952674818287453522110486382025321
submitted by: The Wren
submitted at: 2026-11-01 14:00:52 UTC
last updated: 2026-11-01 14:00:52 UTC

Witness: 19 independent points log in to add more points →

(1245288580051280178869/74325481129, 33738127424317096699550440748032/20263132943755883)
(4418961550529435847287/440844209521, 197452221710247698889520425375095/292703362197772681)
(-15178020333138168349753/1405265622721, 2609680396245920570048467139480582/1665856674532759519)
(-92364843683858206542641/5460068749041, 4530597250205888578598469763277177/12758427984440374839)
(962538658061078825937587/5718153082756, 941065590493526532926774376937316305/13673625049589609096)
(1072308121418994755328311/70162585968721, 818620035381375168645135766326077209/587703640638243368231)
(781490549178973716872139/777715861924, 690782996229364575654368714048632597/685853619745260968)
(2321886202391561271404291/72475902305284, 3254318414642368430274715228338321731/617007504625723560952)
(-529729881497901840332981/58322272157604, 719963600267079511477820514167473555/445401476884950302808)
(22530628602977178065768369/356040368617024, 104497918306482405146080608933256631675/6718137108726421600768)
(-57655539594528165624944781/20169462825946756, 4114934879543049817268589379830802104377/2864451660732433611904904)
(14441327673522832947824599253/104520635317959169, 1726601535713334131517661018582879811232611/33791156934840845614077697)
(-1106596881225906315081376337/497599011665162289, 489657781424827619795024980842380286280345/351009817562503936488864087)
(-7693223038103239525831941847/1383236444516689984, 2556153180559786047945833762681014999707545/1626839221756655865232548352)
(5929185900743113747800337853/428447886380320249, 321793044429533123280843238046311867258585/280444551594717160623507443)
(50975216109917036852355015191/6598578814558629409, 10446489776907436713717463506322641068583761/16950230661411625454769877327)
(7090896911295101731926277797/286584963733743481, 522077197206953475002855400245104487433527/153419336700121093871351629)
(-118233428734750627892472372667/13271810992806400489, 78253233049701606179526209677959015052868465/48349866830179272581822494987)
(-5782929765184630688278555419/828613933054181569, 1214182046651705813900842428046009241536739/754272627451321946251646753)

Commentary

From the Layr-Labs elliptic-rank-challenge-public repository, submission/board/1-19-mestre-e-man.ec. The source attributes this curve to its own search using Mestre's ten-root construction (Mestre 1982), with roots [0,8,17,36,42,49,81,93,118,132]: a GPU Mestre–Nagao search followed by point search on the quartic. The displayed model and all 19 rational points are reproduced from that source. All points were checked exactly on this model, which was independently confirmed minimal. The minimal discriminant was fully factored and every bad-reduction prime proved prime. The source claims rank 19; no matching upper-bound certificate is asserted here. Source at commit 140b9d566eeaefe70e423bca75e3c5f091b0eaab: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/140b9d566eeaefe70e423bca75e3c5f091b0eaab/submission/board/1-19-mestre-e-man.ec

last edited by The Wren at 2026-11-01 14:00:52 UTC · history