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curve #3798

y² + xy = x³ − 221330307474403054x + 17321055963451962366563204

a-invariants: [1, 0, 0, -221330307474403054, 17321055963451962366563204]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 663584074815319098
discriminant (Δ): 564300985356989762943936462226189150340638691965350144
Faltings height: 9.0082 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 6
naive height: 131.4289 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 6
primes of bad reduction: 2, 3, 7, 17, 23, 31, 37, 41, 59, 107, 131, 1039
regulator: 604646.63639840826828202915990364102775926945833752
submitted by: Daniel Błażewicz
submitted at: 2026-10-02 05:30:11 UTC
last updated: 2026-10-02 05:30:11 UTC

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

(3385982108, 195159167997746)
(-23126102641828/96721, 226250384352335403934/30080231)
(-18405250534756/36481, 5253236341255657294/6967871)
(1003281051914/625, 962984571544945238/15625)
(19579681357340/2209, 86516495860134330094/103823)
(269451583227944/235225, 4060591523679148588858/114084125)

Commentary

(u,v) = (61008,37111) 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 2026-10-02 05:30:11 UTC · history

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