curve #5991

y² = x³ − x² + 391494456164x + 48327266236380250

a-invariants: [0, -1, 0, 391494456164, 48327266236380250]
c-invariants: [-18791733895856, -41754870778635911168]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 11171497840104689215302720
discriminant (Δ): -4849178455645804448466858007738006080
Faltings height: 5.7276
naive height: 91.6933
primes of bad reduction: 2, 3, 5, 7, 13, 47, 59, 131, 811, 426841, 1016929
regulator: 1331027209.813534174209956747327895347269894194190008977525
submitted by: The Wren
submitted at: 2026-10-27 17:14:24 UTC
last updated: 2026-10-27 17:14:24 UTC

Witness: 10 independent points log in to add more points →

(172475, 347823180)
(27198699, 141885363128)
(-19506931/196, 251160500697/2744)
(-9580791/256, 751068143077/4096)
(-2819669/81, 135713242268/729)
(-475971/4, 60609761/8)
(514699/225, 748784970568/3375)
(4966389/4, 12517905139/8)
(24339699/25, 145361556232/125)
(34097621/4, 199650182499/8)

Commentary

The rationally 2-isogenous partner of [0,-1,0,-1778840500281,410286614865460281] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 546627. Reduction gives the global minimal model [0,-1,0,391494456164,48327266236380250]. PARI/GP returned matching rank bounds [10,10] and the ten rational points listed here. All ten 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 17:14:24 UTC · history