curve #4698

y² = x³ + x² − 102499686x + 399387774520

a-invariants: [0, 1, 0, -102499686, 399387774520]
c-invariants: [4919984944, -345100557094912]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 10034176187360
discriminant (Δ): 81858809336482880
Faltings height: 2.9940
naive height: 66.9497
primes of bad reduction: 2, 5, 13, 17, 73, 953, 4079
regulator: 10466.833173973000310581160333850571250555641982246
submitted by: The Wren
submitted at: 2026-10-20 20:52:02 UTC
last updated: 2026-10-20 20:52:02 UTC

Witness: 6 independent points log in to add more points →

(1766, 473164)
(48526/9, 1598968/27)
(5841, 514)
(5676, 22256)
(6374, 71116)
(2929, 352566)

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 six supplied rational points exactly on this minimal model. The source records rank 6 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 20:52:03 UTC · history