curve #3966

y² = x³ − 48219165139657693891332x + 97323785926993058356161062184256

a-invariants: [0, 0, 0, -48219165139657693891332, 97323785926993058356161062184256]
c-invariants: [2314519926703569306783936, -84087751040922002419723157727197184]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3407621442773364069580883520
discriminant (Δ): 7171191124534007626574866742993557561635141662713075750595801664000000
Faltings height: 12.0935
naive height: 168.3037
primes of bad reduction: 2, 3, 5, 17, 23, 37, 59, 73, 83, 89, 139, 167, 239, 563, 823
regulator: 114674744390.52101284866428797149433628266018174618517056
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:35:52 UTC
last updated: 2026-10-05 04:35:52 UTC

Witness: 9 independent points log in to add more points →

(7778220152073/4, -21554422100826575675/8)
(46215881954799818/146689, -7146196923209551062174000/56181887)
(391267806122, 202805570384094240)
(-564417356679146551/20866624, 3543248810948375708302703355/95318738432)
(-874120454600008262/4173849, 271030630181083654971590000/8527173507)
(-332929719767798/1521, -760812727187362420000/59319)
(77587117728321313/338724, 6449585674627069310955025/197137368)
(1210020727536705/256, -42045569327579962089569/4096)
(-2794014736403045743/1193287936, -597651493164215771760537983175/41220938461184)

Commentary

Unlisted rational-isogeny neighbor of curve#3676 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 9: 9 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:35:52 UTC · history