curve #3048

y² = x³ − x² − 866893152450363503763740085700x + 61220734062068506723288644020689511073795652

a-invariants: [0, -1, 0, -866893152450363503763740085700, 61220734062068506723288644020689511073795652]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1706951126476128669388929683160
discriminant (Δ): 40075173819244977063109444068398435816801715759648817230455414830191768100293683644217760000
Faltings height: 16.2673
naive height: 218.4177
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 43, 47, 73, 89, 113, 223, 761, 2969
regulator: 1290756.44416902203971720599214457641890155388609825901
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:51 UTC
last updated: 2026-09-26 20:34:51 UTC

Witness: 8 independent points log in to add more points →

(44067817038164, 4806685314955215492570)
(-10736855835311, 8398047577801604981580)
(-130593986210846, 13122671738229868426920)
(960440019693249, 10703970375598940056760)
(54120967930564, 3802914955516162348170)
(-820100918833816, 14852239103603527475850)
(-6811698256976, 8193011203460150583390)
(15735458850860514, 1970429719752218443669720)

Commentary

Dujella - Eroshkin (2008). Source: https://dujella.github.io/tors/z2z4old678.html (curve entry 4). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:34:51 UTC · history