curve #3063

y² = x³ + x² − 8665851076482178429252004x + 9394159653904223016884439949397508576

a-invariants: [0, 1, 0, -8665851076482178429252004, 9394159653904223016884439949397508576]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1605314370710886136520856 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 8
discriminant (Δ): 3525767058015206179396325703699482482391088699992696923470139550411795560704 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 8
Faltings height: 13.2556 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 8
naive height: 183.8779 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 8
primes of bad reduction: 2, 3, 7, 13, 29, 31, 47, 151, 181, 313, 661, 1009, 3049
regulator: 3188682.04724334338008039649722341830284779104798951374
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:32 UTC
last updated: 2026-09-26 20:35:32 UTC

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

(-888210146138, 4048522222598440290)
(1377082663096, 268339815196759512)
(1232801012476, 764525081785096632)
(-1172604283622, 4235970008434474590)
(-387325528466, 3562661647340056170)
(3694116400735, 5271929598002127606)
(26091464591596, 132459174466109758512)
(614639905276, 2073639745189571832)

Commentary

Klagsbrun (2020). Source: https://dujella.github.io/tors/z2z4old678.html (curve entry 23). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:35:32 UTC · history