curve #4580

y² + xy = x³ + x² − 4144842297514034281902042x − 61025416036652507316442748053282620

a-invariants: [1, 1, 0, -4144842297514034281902042, -61025416036652507316442748053282620]
c-invariants: [198952430280673645531298041, 52725959454175623094301481976551448435]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 92600993893266656474637251754052208517553081703634
discriminant (Δ): 4555653297838626585714195878250512679535322497859076881889458254814803682832 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
Faltings height: 13.2070 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
naive height: 181.6653 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
primes of bad reduction: 2, 3, 19, 71, 73, 97, 127, 149, 151, 157, 199, 271, 463, 709, 941, 1217, 1423, 1783, 2087, 4481, 7487
regulator: 122677249080005550883.72982551460608399809426506982685409997798006134
submitted by: jesper-petersen
submitted at: 2026-10-18 08:11:02 UTC
last updated: 2026-10-18 08:11:02 UTC

Witness: 15 independent points log in to add more points →

(-93196970862624/49, 332891096336466613506/343)
(-1507846474944, 1661481777063001230)
(-306633556224, 1086780426367659534)
(-32697549684, -272884231997231526)
(-698035133556, -1578638426784504966)
(-948990585716, -1737165605610808166)
(-1159739621504, -1784955240668783246)
(2156757799509, -1015844216878068093)
(185035569243788325429/64176121, 1777760480292471817275369190752/514114905331)
(5607058386141, 12368485064031055299)
(147030462039301/25, 1672320878758602307511/125)
(-103326359706176/441, -8772319178101836535478/9261)
(-2659365551505339/41209, 3798562490655837290778393/8365427)
(788045658287337277411133421/46536968883601, 21961530377988732381436581318252131901600/317465940867118185401)
(851249278022292683413309/161947490329, 723906633703005339578045677982233000/65172042690628483)

Commentary

Curve and independent-point witnesses from the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r15-z2k3.ec. Specialization u = 12/5, t = -159/38 of the Elkies–Klagsbrun Z/2Z family (arXiv:2003.00077). The original submission describes Mestre–Nagao sieving followed by PARI/GP rank calculations. Submitted to ICARM to document these improvements and make the curve available to other researchers.

last edited by jesper-petersen at 2026-10-18 08:11:02 UTC · history