curve #3632

y² + xy + y = x³ − x² − 233636783676286373627x + 894153434637738073293054084075

a-invariants: [1, -1, 1, -233636783676286373627, 894153434637738073293054084075]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 3744534523427843486091406117969518
discriminant (Δ): 470824754194661964974014144484184477065358692591847532708380672
Faltings height: 10.7284
naive height: 152.3145
primes of bad reduction: 2, 3, 11, 29, 31, 37, 163, 167, 199, 491, 1237, 2063, 3137, 4933, 5413
regulator: 175859167782.97162829868207457612908104326399647874335249846373
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:23:22 UTC
last updated: 2026-09-29 19:23:22 UTC

Witness: 12 independent points log in to add more points →

(-8894264653, 1506183000190362)
(-4166105965, 1339852921415634)
(37038417131, 6561350226610266)
(63889280027, 15708344574285450)
(503428604555, 357032521890142554)
(-107391107628757/9409, 1314358402215245342106/912673)
(-1483401773317/121, 1842313037670777942/1331)
(-1245788111905/169, 3270402467580976038/2197)
(-148758444221/9, 13234350457065854/27)
(-12921500693/9, 29903436912618814/27)
(14005285583675/529, 44314556597849383350/12167)
(25606842889349/1225, 97244474520121145412/42875)

Commentary

Specialization at t = -55/109, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:23:22 UTC · history