curve #409

y² + xy = x³ − x² − 444676196280094399128563694x + 3297768615148916483027180606633461900200

a-invariants: [1, -1, 0, -444676196280094399128563694, 3297768615148916483027180606633461900200]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1809463015179380326823429025654865736412846312311714243982491941376610
naive height: 195.6918
Faltings height: 14.2741
discriminant (Δ): 929329653164922471094982358455833192285812408593237768025722638184893743642937500
primes of bad reduction: 2, 3, 5, 13, 23, 29, 47, 373, 36155150945461, 3658146940752362853285480075388737842001390989
regulator: 81900342214563417833.543127474364635229281229339928789603160691714507845811
submitted by: Bhavik Mehta
submitted at: 2026-08-28 20:29:24 UTC
last updated: 2026-08-28 20:29:24 UTC

Witness: 18 independent points

(-23821969611234, 19292279201517010092)
(193596845456411541/12769, -9238406465636915675074902/1442897)
(-34733987926513258649/3538161, 545450958953350121329537546772/6655280841)
(1819330663103556/121, 4399702735070284506012/1331)
(200378240327380955859/3829849, -2631308118957295461517157749719/7495014493)
(2956322830396491901/63001, 4630831126884244273150923022/15813251)
(1583311003995846510159/78198649, -35224420629098416522803678391956/691510653107)
(-993421639266789/961, -1825860788461954312210173/29791)
(427677231953043451746/22174681, -4546388911536769746155696630082/104420572829)
(-21103176768966, 57303364409961390054)
(477019281514227331969/2985984, -10331554847855041292836560929441/5159780352)
(-186145719720300831/142129, 3336745316937228619070544246/53582633)
(2248390547695411573281/47665216, -97220356584163634901858190965237/329080651264)
(8075162436474904855494/17952169, -724864847724571257861062902779234/76063340053)
(5765547517132050, -437782034687446204684200)
(22793168168284175185254/149255089, -3409820466005214264595759014424044/1823449422313)
(237208828931867456919/7579009, 2953636698279764083383192264309/20865011777)
(435507017681691, 9078032059655153293917)

Commentary

Rank-18 curve from Mestre's quartic construction, specialized at T = 2539/5. 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 20:29:24 UTC · history