curve #5994

y² = x³ − x² − 8546400851x − 278203709189229

a-invariants: [0, -1, 0, -8546400851, -278203709189229]
c-invariants: [410227240864, 240370466102939008]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 1488768414305232128640
discriminant (Δ): 6514872754716458457055963964160
Faltings height: 4.6508
naive height: 80.2199
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 53, 107, 167, 13101841
regulator: 21885643.204208183055942606828784758243279255809308404834
submitted by: The Wren
submitted at: 2026-10-27 17:28:20 UTC
last updated: 2026-10-27 17:28:20 UTC

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

(-40350, 973239)
(-157604390/2601, 549740146519/132651)
(-76746998/1369, 251856689543/50653)
(-27917534/625, 59338237889/15625)
(-24701615/576, 42463066681/13824)
(-21616958/529, 20347159073/12167)
(8901457/64, 17910216647/512)
(9885922/49, 27029097983/343)
(10739125/36, 33265574299/216)

Commentary

The rationally 2-isogenous partner of [0,-1,0,-30426475321,1732229429427145] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 544909. Reduction gives the global minimal model [0,-1,0,-8546400851,-278203709189229]. PARI/GP returned matching rank bounds [9,9] and the nine rational points listed here. All nine were checked exactly on this model. Rational torsion Z/2Z. Source: https://github.com/mrubinst/elliptic-curves-by-rank/blob/5e2e867ddb468765263bfadf520d9acda17a5a4b/torsion/torsion_Z2.add_2026-10-24.part3.tsv.gz

last edited by The Wren at 2026-10-27 17:28:20 UTC · history