curve #4696

y² + xy + y = x³ − 297359779x + 1961923904106

a-invariants: [1, 0, 1, -297359779, 1961923904106]
c-invariants: [14273269369, -1695123663051853]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 543836626686688370
discriminant (Δ): 19904491235494263611488100
Faltings height: 3.6883
naive height: 70.1450
primes of bad reduction: 2, 5, 7, 23, 127, 179, 809, 1289, 14249
regulator: 44672.4455390298993782262476574059289875798634119045904
submitted by: The Wren
submitted at: 2026-10-20 20:48:56 UTC
last updated: 2026-10-20 20:48:56 UTC

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

(-3207, 1699418)
(1089, 1279842)
(10576, -903)
(-19496, 600537)
(2163, 1151678)
(10611, 32046)
(54789, 12225692)
(10934, -138912)

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 20:48:56 UTC · history