curve #1246

y² = x³ − x² − 234752993400x − 37752339354870384

a-invariants: [0, -1, 0, -234752993400, -37752339354870384]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 72554931074823247344
discriminant (Δ): 212261836373132477107527360278341632
Faltings height: 5.5032
naive height: 90.1590
primes of bad reduction: 2, 3, 7, 47, 59, 71, 79, 139, 227, 239, 263
regulator: 8017544.1097938529080858825522747036190702990147302935074
submitted by: jesper-petersen
submitted at: 2026-09-24 14:56:47 UTC
last updated: 2026-09-24 14:56:47 UTC

Witness: 9 independent points log in to add more points →

(2112906, 2983124214)
(2144714, 3053525258)
(11704296, 40007404044)
(-106336014/529, 441022511286/12167)
(-8067984/25, 8288155764/125)
(5305072/9, 4572440684/27)
(6137890/9, 9327553858/27)
(10057729/9, 28260376228/27)
(365768616/625, 2485333294164/15625)

Commentary

The curve was found by adapting Dujella’s 2002 use of Fermigier’s rank-\(\ge 8\) construction with rational 2-torsion. Dujella used products of three primes \(p_i\equiv1\pmod4\), giving four representations as sums of two squares. I instead used products of four such primes, giving eight representations and many possible Fermigier curves to test. he best case came from \(5\cdot13\cdot29\cdot41\), using \((106,257)\), \((113,254)\), \((142,239)\), and (166,223)

last edited by jesper-petersen at 2026-09-24 14:56:47 UTC · history