curve #408

y² + xy = x³ − x² − 276891740847977926726940094x + 1884532832341693905066312410598171652600

a-invariants: [1, -1, 0, -276891740847977926726940094, 1884532832341693905066312410598171652600]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1921048791987278542375790503041156402647469228660080670360277605330
naive height: 194.3921
Faltings height: 14.1461
discriminant (Δ): -175574981472651644515284153152000581141671079737037803387236807293960702438312500
primes of bad reduction: 2, 3, 5, 7, 13, 43, 4987, 18859754273231, 57997682476170649245198004209712540060550417
regulator: 21879331590853938884.291277946765253852174072798582542275279886881693420907
submitted by: Bhavik Mehta
submitted at: 2026-08-28 20:29:18 UTC
last updated: 2026-08-28 20:29:18 UTC

Witness: 18 independent points

(-8027583276034, -59916521839841622508)
(110579138173309526/12769, -16833976176009319948919522/1442897)
(-39275422713709606494/2042041, -28471732838972902310566194362/2918076589)
(799681980470396/121, -24659002233759761630168/1331)
(15204738199492180121/654481, -47329151186729263651405537007/529475129)
(-596097294527084154/63001, -956322865641440068604067718/15813251)
(666957054379235011154/56055169, -6952319984060723792978051121374/419685050303)
(-18587941837203319/961, -58330309436108019299038/29791)
(185217128559293107676/16248961, -947742666045528038062515348728/65499561791)
(-419780228754754/529, 558060458721497985538348/12167)
(60423170662936046241/564001, -464347597727089192165161754358/423564751)
(5833837112380376983094/349577809, 285747900608833012595004204442766/6536056294873)
(4320424875300107714/552049, 5755235284857417756797778784/410172407)
(3648287888066411002799/83302129, 206490172176042496998518213697956/760298531383)
(18954401339362921599821/254689681, 2549593455208401910957316556125143/4064592619079)
(97310969184818314979/2365444, 891996379417223722451927434549/3638052872)
(10321488018550628245151/1061521561, 365194897622623942556799086317697/34585433978941)
(109680229741693103029/11370384, 404220412258421818055999737361/38340934848)

Commentary

Rank-18 curve from Mestre's quartic construction, specialized at T = 1861/5. The sextuple is [0, 113, 550, 753, 868, 1058], the canonical primitive representative from u = -3, v = -1/2 in the Mestre-Fermigier family, whose generic member has rank 12.

last edited by Bhavik Mehta at 2026-08-28 20:29:18 UTC · history