curve #4193

y² = x³ − 1693886074426372525887x + 24856451483538231050129831591170

a-invariants: [0, 0, 0, -1693886074426372525887, 24856451483538231050129831591170]
c-invariants: [81306531572465881242576, -21475974081777031627312174494770880]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 16727690052741601039120629516045392048387642326599525010968
discriminant (Δ): 44143437298542131612581150537590891073740278391925856930543937792
Faltings height: 11.1467
naive height: 158.2575
primes of bad reduction: 2, 3, 7, 11, 17, 144981901, 48511425331, 171842641613, 48950125555422649693799
regulator: 57045721336677772.164147602315923431764054067867067913400278010908392793731
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-12 04:10:17 UTC
last updated: 2026-10-12 04:10:17 UTC

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

(-1937770540187659/241081, -729228678078715512198798/118370771)
(46826439778337/49, 320139873797186393742/343)
(-464077505655639/58564, -87113753550275506819933/14172488)
(-38015202003, -5857643839199702)
(3966036693977/256, -6255704023786339683/4096)
(-141282972491093/3969, -1582382897791898520032/250047)
(18278551835, -40385480800920)
(-24903800240952537/737881, -4184351182803945334839224/633839779)
(360518603815/9, -124552816313710400/27)
(-16836223248042/529, -82993756697801919284/12167)
(200560085013857/961, -2788496862357887367918/29791)
(969904941722, -954351430695210702)
(-716481906953338/39601, -55490121394639847082552/7880599)
(-1069634837200604299/80694289, -4861569708671506896767519202/724876798087)
(23110412809931/2209, -298754305107375421392/103823)
(-36380946623837/4761, -2007778653622532274904/328509)
(-5087789795233239/303601, -1165438315177648174142438/167284151)
(30016824379421/2209, -216496465017366751098/103823)

Commentary

Mestre rank-12 quartic family on locus sextuple [0,14,790,819,859,920] at T=491 (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 04:10:17 UTC · history