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curve #5670

y² = x³ − 188588857x

a-invariants: [0, 0, 0, -188588857, 0]
c-invariants: [9052265136, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1138104223506126368
discriminant (Δ): 429267549315785808477362752
Faltings height: 3.7998
naive height: 68.7788
primes of bad reduction: 2, 73, 109, 137, 173
regulator: 121273.5662152030865262602076768471977034862161183862
submitted by: Michael Rubinstein
submitted at: 2026-10-24 04:20:40 UTC
last updated: 2026-10-24 04:20:40 UTC

Witness: 7 independent points log in to add more points →

(-12629, 606192)
(1730173/9, 2269986976/27)
(-173, 180612)
(-1730173/169, 2034683448/2197)
(113661, 38038548)
(62453, 15225384)
(14013, 330084)

Commentary

Rank 7, proved: 7 independent points listed, matching the 2-descent upper bound. Torsion Z/2. Member of the family y^2 = x^3 + a x.

last edited by Michael Rubinstein at 2026-10-24 04:20:40 UTC · history

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