curve #457

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y² = x³ + x² − 274772680388272620238822915925029108x + 55109431017839872572667629268320431851587994748732288

a-invariants

[0, 1, 0, -274772680388272620238822915925029108, 55109431017839872572667629268320431851587994748732288]

rank (lower bound)

≥ 21

torsion subgroup

trivial

conductor (N)

4985496876385205151318529244550971267815315488413817176648366062065676226102431879800

naive height

256.4174

Faltings height

19.2110

discriminant (Δ)

15696711652466593784984814071054398908676862294563966753627506827350997270194814689646347027702776516000000

primes of bad reduction

2, 3, 5, 7, 13, 17, 19, 29, 293, 358987, 483914399, 208301526751570139, 131343899330485327769779455587343605984077

regulator

7831985639918254024867.9835468280953149474752781845151313384463365264286116879476

submitted by

Bhavik Mehta

submitted at

2026-08-30 15:55:30 UTC

later contributions

rank ≥ 20 → ≥ 21 · Jason Ma · 2026-09-03 16:32:30 UTC

Witness: 21 independent points log in to add more points →

(-520813263718905358, -238633475489700629811325398)
(-355964894258908576, -328350710833589068423906464)
(8453597435259125858, 24532719835613092176969017850)
(-484765993908264892, -272746787538496538994224400)
(3009238818873955042/9, 670526957497553809427961550/27)
(171037087191806258, 114527626491311181570230250)
(458785078026231938, -160045706200037599657129050)
(-2200100045164934443/4, 1596868512997713982884318825/8)
(115875145966645272608, -1247328958626610811314566075600)
(597244387722017141/4, -1055622143996426584998146727/8)
(1337485094080435778, -1442285402105459163074391450)
(1279136750591221178, 1340354212476561094926794850)
(37190151245892097013/4, -226446455320629343268220917529/8)
(528760140685323983, 240114978204363129039340350)
(-296001634134098506, 332427176832647198134399266)
(345196676059468214, -37319006186174989582296354)
(181525958194605278, 105889758856010865796747050)
(515781718147924, -234451932024163349883754536)
(-2932477948987312, -236463888630870654222583800)
(-3490491421050016, -236787837894951553551571776)
(-69893442383372243/4, -1958044709714095270442559525/8)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -31/18. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:55:30 UTC · history