curve #3972

y² + xy = x³ − 656136142870032910x − 109429194175305294926471164

a-invariants: [1, 0, 0, -656136142870032910, -109429194175305294926471164]
c-invariants: [31494534857761579681, 94546823720221972529828716175]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 58287634249314930726735678
discriminant (Δ): 12905386333681135856517646882479550937055476765118569472
Faltings height: 9.2721
naive height: 134.6890
primes of bad reduction: 2, 3, 11, 19, 31, 67, 83, 101, 107, 163, 199, 211, 283, 1171
regulator: 2015294999873.294394557022458575689178141604534440680310125
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:37:03 UTC
last updated: 2026-10-05 04:37:03 UTC

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

(-491154214, 9713766935648)
(-22427769230194/32041, -15052297725007339646/5735339)
(7785084920684/2809, -20715039164829785798/148877)
(119612007239/100, -28584615335599483/1000)
(32896234844/25, 4516440483365818/125)
(-15777657964/25, 912589643223394/125)
(-255966851637608/1292769, 5243056057286125938562/1469878353)
(-391880094530221/2220100, -3118115983812042339713/3307949000)
(6749479935587666/6754801, -267749538154405062108136/17555727799)
(24892807799998496/7722841, -3794811227902199615959078/21461775139)

Commentary

Unlisted rational-isogeny neighbor of curve#3619 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 10: 10 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:37:03 UTC · history