curve #3145

y² = x³ − 315631503507622339923146392582227x + 2072228053755856864158196203880686412361893003346

a-invariants: [0, 0, 0, -315631503507622339923146392582227, 2072228053755856864158196203880686412361893003346]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 9956040340114873727319206638572191492798131920
discriminant (Δ): 157367260839641970679721464950952395993803984657681836096326623448934721051892006075119254448640000
Faltings height: 17.6044
naive height: 236.1100
primes of bad reduction: 2, 3, 5, 7, 17, 29, 37, 41, 43, 127, 151, 199, 211, 367, 769, 1153, 3019, 13873, 23197, 34471
regulator: 55930928709.8170389048083084304911544109937907721999994551936
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:27 UTC
last updated: 2026-09-26 20:39:27 UTC

Witness: 11 independent points log in to add more points →

(-20230370204964983, -421816927565146336143480)
(-18385457713072919, -1288608689198829837436920)
(15615691751642497, -975349962521333965224000)
(301276352453444473/4, -161112473370726659001899595/8)
(33000572294573513/4, 1380268510109424490727725/8)
(6179239441242217, 598169974517352025641720)
(41142281826465527544193/2879809, 3381886289254955378457102713670240/4887035873)
(299042790321073687, -163248467182728124569159690)
(6486704984610865, 545676559760580599572704)
(204594166513207396346233/24039409, 5800730753279830273971505975698120/117865222327)
(-548808587980476047/64, 1042804094168807838620125065/512)

Commentary

Dujella - Kulesz (2006). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 11). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:39:27 UTC · history