curve #5998

y² = x³ + x² − 5110853119x − 67393864955439

a-invariants: [0, 1, 0, -5110853119, -67393864955439]
c-invariants: [245320949728, 58226827395800960]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 934186815498928733056
discriminant (Δ): 6581963158660205383792705750784
Faltings height: 4.6090
naive height: 78.6775
primes of bad reduction: 2, 7, 41, 113, 127, 503, 953, 32713
regulator: 261663155.83828788720749693067172308514394958770374314889
submitted by: The Wren
submitted at: 2026-10-27 18:04:24 UTC
last updated: 2026-10-27 18:04:24 UTC

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

(-58455, 5623566)
(-14816, 2253095)
(79984, 5959775)
(92368, 15767215)
(126409, 36145310)
(233368, 107001485)
(-1030424/25, 1069759999/125)
(3784573/36, 5098514795/216)
(559551856/225, 13230617880079/3375)

Commentary

The rationally 2-isogenous partner of [0,1,0,-40339155689,3072117231780391] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 544799. Reduction gives the global minimal model [0,1,0,-5110853119,-67393864955439]. 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 18:04:24 UTC · history