curve #4681

y² + xy + y = x³ − 962577656x + 11494564154358

a-invariants: [1, 0, 1, -962577656, 11494564154358]
c-invariants: [46203727465, -9931372734956725]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 137771321171561050
discriminant (Δ): 1639858970788010003498000
Faltings height: 3.7774
naive height: 73.6690
primes of bad reduction: 2, 5, 29, 31, 109, 149, 281, 331, 2029
regulator: 56470.1853054129751567344036682235203130011036824017279
submitted by: The Wren
submitted at: 2026-10-20 20:33:43 UTC
last updated: 2026-10-20 20:33:43 UTC

Witness: 8 independent points log in to add more points →

(13362, 1002446)
(17277, 137231)
(18321, 85031)
(122317/9, -26119987/27)
(24477, 1599731)
(209837, 95021171)
(109408/9, 33876742/27)
(7596, 2145881)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, torsion/torsion_Z2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. I checked the eight supplied rational points exactly on this minimal model. The source records rank 8 with independent points and a matching 2-descent upper bound. Rational torsion Z/2Z. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 20:33:43 UTC · history