curve #4680

y² + xy + y = x³ − 34654753x + 78518579712

a-invariants: [1, 0, 1, -34654753, 78518579712]
c-invariants: [1663428121, -67842548013565]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 282151265615194
discriminant (Δ): 47936935725490230212
Faltings height: 2.9335
naive height: 63.6964
primes of bad reduction: 2, 17, 19, 137, 263, 761, 937
regulator: 6097.528111644173670407888182209128252012766313577418
submitted by: The Wren
submitted at: 2026-10-20 20:31:58 UTC
last updated: 2026-10-20 20:31:58 UTC

Witness: 7 independent points log in to add more points →

(2761, 60958)
(3388, -1020)
(4053, 66126)
(3441, 2478)
(-1327, 350190)
(40657/9, -3274070/27)
(-6587, -141620)

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 seven supplied rational points exactly on this minimal model. The source records rank 7 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 20:31:58 UTC · history