curve #3965

y² + xy + y = x³ − x² − 62721061370796884284301x − 2761181947863191529851422253557579

a-invariants: [1, -1, 1, -62721061370796884284301, -2761181947863191529851422253557579]
c-invariants: [3010610945798250445646433, 2385661202967345231047720954079156975]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4145713793217534004171727022
discriminant (Δ): 12497760813342119729665870577812294292331107581497731870516243966459904
Faltings height: 12.1452
naive height: 169.0926
primes of bad reduction: 2, 3, 7, 11, 13, 19, 37, 47, 59, 73, 107, 827, 1031, 3517, 5039
regulator: 73151550727.160258127226369828350669856114347752937999070
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:35:42 UTC
last updated: 2026-10-05 04:35:42 UTC

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

(1878030396831, -2550149274655697240)
(785142030767296719/1181569, 640853133847148687304006664/1284365503)
(3509707282776921/3136, -202441898776295493643777/175616)
(50047231437579/169, -149577902725131613244/2197)
(1123416623636621211/534361, 1182067095236519443910341160/390617891)
(7705687900878679/841, 676167010224193683938120/24389)
(32370128042397231/58081, 5150554722937941040675960/13997521)
(51532704806791791/185761, 2760122560674501238657160/80062991)
(1519280695614890999919/53919649, -59216111846654864855083442017144/395931982607)

Commentary

Unlisted rational-isogeny neighbor of curve#3679 (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 04:35:42 UTC · history