curve #60

y² + xy = x³ − x² − 1536664x + 648294124

a-invariants: [1, -1, 0, -1536664, 648294124]
rank (lower bound): ≥ 10
conductor (N): 25440555737235843986
naive height: 54.3490 ★ record for rank ≥ 10
Faltings height: 2.5105 ★ record for rank ≥ 10
discriminant (Δ): 50881111474471687972 ★ record for rank ≥ 10
primes of bad reduction: 2, 53993, 235591240876001
regulator: 488600.0685314399839133782564429436509012776371221864320710
submitted by: David Renshaw
last updated: 2026-07-01 22:49:33

Witness: 10 independent points

(-1412, 1890)
(-1411, 3123)
(-1386, 11378)
(-1372, 13822)
(-1353, 16463)
(-1350, 16832)
(-1329, 19154)
(-1274, 23810)
(-1259, 24840)
(-1237, 26215)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=10: [1,-1,0,-1536664,648294124], |Delta|=50881111474471687972; their I-column records 207 integral x-coordinates. Also appears in Table 2 with conductor N=25440555737235843986 and |Delta|/N=2. The submitted points certify rank >= 10 here.

last edited by David Renshaw at 2026-05-29 00:13:07 · history