curve #4117

y² + xy + y = x³ + x² − 174487014x + 387295820946

a-invariants: [1, 1, 1, -174487014, 387295820946]
c-invariants: [8375376673, -334686404622545]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/4ℤ
conductor (N): 543416329599
discriminant (Δ): 275168986539125259851015889
Faltings height: 3.7676
naive height: 68.5457 ☆ record for ℤ/4ℤ torsion, rank ≥ 5
primes of bad reduction: 3, 11, 13, 17, 37, 101, 127, 157
regulator: 5016.41997726863319950311363982733237844220774661
submitted by: NDElkies
submitted at: 2026-10-09 15:33:10 UTC
last updated: 2026-10-09 15:33:10 UTC

Witness: 5 independent points log in to add more points →

(1815, 275832)
(1875411/25, -2534070374/125)
(-10810, -999798)
(215387, -99882450)
(14219, -891030)

Commentary

Smallest naïve height H of the 5 curves with H < 2^100 and this MW group. This curve is [a,b,a*b,0,0] with (a,b)=(127, -15857). Curves 2-5 are at (a,b)=(265,-2959) [curve#4114], (80, 81205/4) [curve#1685], (165, 16215) [curve#1652], and (288, 60409/4).

last edited by NDElkies at 2026-10-09 15:58:07 UTC · history