curve #3964

y² = x³ − 86874247857446531255172x + 4185024865555893258206024187372736

a-invariants: [0, 0, 0, -86874247857446531255172, 4185024865555893258206024187372736]
c-invariants: [4169963897157433500248256, -3615861483840291775090004897890043904]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3716012842930786646909643840
discriminant (Δ): 34395471863326763409908678779571360723610636206386854460371585600000000
Faltings height: 12.2287
naive height: 170.0698
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 37, 59, 109, 137, 157, 163, 199, 277, 449
regulator: 134516945192.29146996902811550424130951477269473974684480
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:35:31 UTC
last updated: 2026-10-05 04:35:31 UTC

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

(-61426775512903/196, 68819495171732140275/2744)
(19812449283517073/432964, -4979281222923448068646425/284890312)
(58135146620155817/27556, -13882783458958932787771275/4574296)
(55310639175297/196, -127002842165180805025/2744)
(2834402467553/4, 4370261396619524175/8)
(-9904282237710255591/41731600, 28827965111316387116482206077/269586136000)
(-698931050799211047/3247204, 664909847064401514134684675/5851461608)
(244126245854513/784, -1878900765937051638825/21952)
(317336215795647647921/8179600, 5652843206201839746772956804969/23393656000)

Commentary

Unlisted rational-isogeny neighbor of curve#3678 (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 04:35:32 UTC · history