curve #5995

y² + xy + y = x³ − x² + 2405340139x + 202772063451260

a-invariants: [1, -1, 1, 2405340139, 202772063451260]
c-invariants: [-115456326687, -175195582375358961]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 80438488494668794269
discriminant (Δ): -18653092465808033563691792211183
Faltings height: 4.6812
naive height: 79.4094
primes of bad reduction: 3, 13, 61, 131, 179, 367, 1093, 1511
regulator: 916999.02138933375736162488281155479923445459497026926410
submitted by: The Wren
submitted at: 2026-10-27 17:33:23 UTC
last updated: 2026-10-27 17:33:23 UTC

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

(-178293/4, 21343913/8)
(-38262, 7416631)
(-32187, 9608005)
(-31086, 9912982)
(19887/4, 117243215/8)
(73876, 27956904)
(83418, 31325230)
(156084, 66109123)
(1113954, 1176379543)

Commentary

The rationally 2-isogenous partner of [1,-1,1,-40537739066,2932199823257696] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 544601. Reduction gives the global minimal model [1,-1,1,2405340139,202772063451260]. 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 17:33:23 UTC · history