curve #3375

y² + xy = x³ − 250132524775x − 31792439260195159

a-invariants: [1, 0, 0, -250132524775, -31792439260195159]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/10ℤ
conductor (N): 34172754
discriminant (Δ): 564943741174552065488203015145378592
Faltings height: 5.5633
naive height: 90.3494
primes of bad reduction: 2, 3, 7, 11, 17, 19, 229
regulator: 28.5297547150703122628590221330616137893493
submitted by: David Renshaw
submitted at: 2026-09-26 21:16:51 UTC
last updated: 2026-09-26 21:16:51 UTC

Witness: 2 independent points log in to add more points →

(606128, 197892269)
(2837244536/2209, 137896926128773/103823)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/10ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with d=t^2/(t-(t-1)^2), c=td-t, b=cd, at t = 11/28; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 34172754, torsion ℤ/10ℤ.

last edited by David Renshaw at 2026-09-26 21:16:51 UTC · history