curve #1210

y² + xy + y = x³ + x² − 223521435x + 1911696935401

a-invariants: [1, 1, 1, -223521435, 1911696935401]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/5ℤ
conductor (N): 2462003614
discriminant (Δ): -864212044892876703988579328
Faltings height: 3.8688
naive height: 70.0813 ☆ record for ℤ/5ℤ torsion, rank ≥ 4
primes of bad reduction: 2, 7, 17, 43, 240571
regulator: 97.51278782964378544263322840471627982471037234
submitted by: David Renshaw
submitted at: 2026-09-24 13:01:15 UTC
last updated: 2026-09-24 13:01:15 UTC

Witness: 4 independent points log in to add more points →

(-10397, 1769254)
(635, 1330102)
(10547, 847718)
(228793, 109097734)

Commentary

Found by an independent rational-parameter search in the Tate family y^2+(1-t)xy-ty=x^3-tx^2 at t=119/172, whose point (0,0) has order 5. Submitted in the global minimal model [1,1,1,-223521435,1911696935401]. PARI/GP ellrank returns bounds [4,4], proving rank exactly 4, and elltors returns invariant factors [5], proving torsion exactly Z/5Z. The four supplied rational points were found by PARI and independently checked by the ICARM exact independence verifier.

last edited by David Renshaw at 2026-09-24 13:01:15 UTC · history