curve #5999

y² = x³ + x² − 6781592505x − 147166792022201

a-invariants: [0, 1, 0, -6781592505, -147166792022201]
c-invariants: [325516440256, 127150155208540160]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 187638029667623179328
discriminant (Δ): 10604670264513178856790892777472
Faltings height: 4.6579
naive height: 79.5260
primes of bad reduction: 2, 41, 47, 79, 83, 113, 131, 193, 241, 337
regulator: 20398305.221673015864112671798449599387022971186163639618
submitted by: The Wren
submitted at: 2026-10-27 18:06:58 UTC
last updated: 2026-10-27 18:06:58 UTC

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

(-62529, 5692336)
(-60225, 6543632)
(-44319, 8144836)
(-37089, 7303264)
(124863, 30867280)
(184631, 69961336)
(10413635, 33603919832)
(-1127789/25, 1022856384/125)
(-358865/9, 208873736/27)

Commentary

Rationally 2-isogenous to the rank-9 curve [0,1,0,-42104490145,3212429325415839] in Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz (recorded source index 544625). The source points were transported through the degree-2 isogeny to the minimal model [0,1,0,-6781592505,-147166792022201]. 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 18:06:58 UTC · history