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curve #5518

y² + xy + y = x³ + x² − 616967544223889415x + 186292884048725070093522372

a-invariants: [1, 1, 1, -616967544223889415, 186292884048725070093522372]
c-invariants: [29614442122746691921, -160957052040206776481403518969]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 40914952311895935
discriminant (Δ): 37698352886350825733869659528435365470643999351390625
Faltings height: 8.9905
naive height: 134.5044
primes of bad reduction: 3, 5, 7, 13, 19, 47, 61, 73, 83, 197, 461
regulator: 9379.1687832597904616222739902573698927114057148925
submitted by: Daniel Błażewicz
submitted at: 2026-10-22 21:22:26 UTC
last updated: 2026-10-22 21:22:26 UTC

Witness: 6 independent points log in to add more points →

(-42763518, 14580775281971)
(6950129543/4, 528231464058483/8)
(-139766337767/169, 25135915689619487/2197)
(53215294697/121, 181252974124401/1331)
(254797267447/196, 109110539295483949/2744)
(115771628338933/374544, 1148826406833927326663/229220928)

Commentary

(u,v) = (10244,74119) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 6.

last edited by Daniel Błażewicz at 2026-10-22 21:22:26 UTC · history

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