curve #4582

y² + xy + y = x³ − 47884844596302273x − 696780484060531382943068

a-invariants: [1, 0, 1, -47884844596302273, -696780484060531382943068]
c-invariants: [2298472540622509081, 602018334780590303929046915]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 1470210055845288515574478669853106
discriminant (Δ): 6817331735714869117536734578049899026522846195750672 ☆ record for ℤ/2ℤ torsion, rank ≥ 12
Faltings height: 8.6361 ☆ record for ℤ/2ℤ torsion, rank ≥ 12
naive height: 126.8363 ☆ record for ℤ/2ℤ torsion, rank ≥ 12
primes of bad reduction: 2, 3, 19, 43, 53, 71, 101, 109, 193, 227, 331, 997, 3391, 4339, 34033
regulator: 84995896596978.473008400051111674161189659311348740527388654478
submitted by: jesper-petersen
submitted at: 2026-10-18 08:12:21 UTC
last updated: 2026-10-18 08:12:21 UTC

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

(-185608537, 1340519298570)
(-105555277, 1783765050246)
(-86583532, 1673413255056)
(-21841372, 581965292736)
(-72806809, -1550326660470)
(-119413369, -1821623281362)
(315229796, -3941327388645)
(264830036, -2279543569317)
(13001661356, 1482295471059288)
(249956380247/529, -110294411004400698/12167)
(23411986473629506316/35884103761, -106570641552948007432177008195/6797561659549991)
(690463992694511/105625, -18133023607123734059466/34328125)

Commentary

Curve and independent-point witnesses from the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r12-z2low.ec. Specialization u = 10/3, t = 1/8 of the Elkies–Klagsbrun Z/2Z family (arXiv:2003.00077). The original submission describes a rational-parameter census using a height filter, Mestre–Nagao sieving and PARI/GP rank computations. Submitted to ICARM to document these improvements and make the curve available to other researchers.

last edited by jesper-petersen at 2026-10-18 08:12:22 UTC · history