curve #55

y² + xy = x³ − 22x + 219

a-invariants: [1, 0, 0, -22, 219]
rank (lower bound): ≥ 5
conductor (N): 20384311
naive height: 24.3180 ★ record for rank ≥ 5
Faltings height: 0.0837 ★ record for rank ≥ 5
discriminant (Δ): -20384311
primes of bad reduction: 3089, 6599
regulator: 23.0216507645579847918634719671775842289367147243
submitted by: David Renshaw
last updated: 2026-07-01 23:01:14

Witness: 5 independent points

(-7, 10)
(-6, 15)
(-5, 17)
(-2, 17)
(-1, 16)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=5: [1,0,0,-22,219], |Delta|=20384311; their I-column records 29 integral x-coordinates. The submitted points certify rank >= 5 here.

last edited by David Renshaw at 2026-05-29 00:13:04 · history