curve #4200

y² = x³ − 1572754513082802069475x + 25243298865338072338192081505650

a-invariants: [0, 0, 0, -1572754513082802069475, 25243298865338072338192081505650]
c-invariants: [75492216627974499334800, -21810210219652094500197958420881600]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 76711644195431677443287364353709744687885843783919989200
discriminant (Δ): -26301783914053199345599188151553303838258792560479997449051520000
Faltings height: 11.1157
naive height: 158.1354
primes of bad reduction: 2, 5, 7, 11, 23, 11525771, 91657416306312217, 14643574507469001823127587
regulator: 14769618606876138.398917328686155972762160818107426084435264955963744503081
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-12 15:12:38 UTC
last updated: 2026-10-12 15:12:38 UTC

Witness: 18 independent points log in to add more points →

(-3457338932283950/152881, 419479869614425008197270/59776471)
(704160568319525/49, 18685550649395726586160/343)
(-21983369718415/961, 209069066633909594420/29791)
(164310673150/9, 43662059186593240/27)
(62082601555, 12918375976212470)
(1954647280909/121, 2679362766332908048/1331)
(13463416501867325/351649, 959014804308362629051840/208527857)
(7682274653570475/477481, 668389601011663290428270/329939371)
(5433282443065/144, 7653492365140681235/1728)
(14841196475, 2273912615018870)
(-35601074725, -6009404920280630)
(271939079745/64, 2210352288610426015/512)
(2296005900475/729, 88728873290807077790/19683)
(860305647425, 797123958533984620)
(1124260375303576725/30880249, 691548770821697914177542160/171601543693)
(-43485562475, 3377050417626880)
(28709623218267117525/2087667481, 237810201980439537133814187520/95387614874371)
(-14281079228247895/326041, 591540601510033950040420/186169411)

Commentary

Mestre rank-12 quartic family on locus sextuple [0,7,1182,1375,1382,1532] at T=391 (y^2=(g^2-p(x-T)p(x+T))/T^2, Jacobian); 18 independent points from a quartic rational-point search.

last edited by Andrii Prokhorenkov at 2026-10-12 15:12:38 UTC · history