curve #4670

y² + xy + y = x³ − x² − 27728393430547348693562462x + 55881924786702165148992915159949517549

a-invariants: [1, -1, 1, -27728393430547348693562462, 55881924786702165148992915159949517549]
c-invariants: [1330962884666272737290998161, -48281983015704681355748880470878573670841]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 267850366109224468929137248470
discriminant (Δ): 15393840473670775536345619115358466634977146502004418756018583783104000000000
Faltings height: 13.4549
naive height: 187.3671
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 31, 43, 59, 113, 131, 2239, 32401, 1213873
regulator: 200926415615.5679519571584663822134835445739622196448177793
submitted by: jesper-petersen
submitted at: 2026-10-20 19:44:01 UTC
last updated: 2026-10-20 19:44:01 UTC

Witness: 10 independent points log in to add more points →

(-6044372458743, 1629392165978923171)
(19243712248857, 81539092004797715971)
(-6032726129661687/2209, 1095021550276840165366733/103823)
(-29199304563503/121, 10527460730005843528601/1331)
(158974705478313/49, 91288649947688335813/343)
(44439094929380433/1369, 9251690289441687577264663/50653)
(303030999696733909/180625, 288096770995119078767554727/76765625)
(54770745510572501121/16933225, 12809154872133130877865568961/69680220875)
(283338413439504673137/87815641, 59073586534654230883187360681/822920371811)
(4017287537690907281793/468419449, 214791503402314532898750271549897/10138002134707)

Commentary

Rank exactly 10 by PARI/GP 2.17.4 ellrank. Constructed by specializing a rank-5 elliptic K3 deformation with rational 3-torsion at w=8/3, t=121/41. Found by precomputed Mestre-Nagao sieving of 27,892,344 rational parameter pairs.

last edited by jesper-petersen at 2026-10-20 19:44:01 UTC · history