curve #5694

y² = x³ + 236073094854743x

a-invariants: [0, 0, 0, 236073094854743, 0]
c-invariants: [-11331508553027664, 0]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1783376195657487548808627073568 ☆ record for quartic twists, rank ≥ 10
discriminant (Δ): -842014275598281539037192974161391420269466048 ☆ record for quartic twists, rank ≥ 10
Faltings height: 7.3098 ☆ record for quartic twists, rank ≥ 10
naive height: 110.8991 ☆ record for quartic twists, rank ≥ 10
primes of bad reduction: 2, 19, 157, 197, 313, 769, 1669
regulator: 25455918228.90717815547251096741520614721668041735827843323
submitted by: Michael Rubinstein
submitted at: 2026-10-24 13:20:26 UTC
last updated: 2026-10-24 13:20:26 UTC

(78591541, 709918476372)
(401723293/3249, 1000577026554616/185193)
(772878067/9, 21827741272532/27)
(926209881/784, 367683402456483/21952)
(3624942825/64, 226136396322165/512)
(8335888929/484, 1019913608450889/10648)
(23280454537/7056, 16918690885812827/592704)
(97343800499/265225, 1271790935852104932/136590875)
(712255398489/8464, 611046992803773219/778688)
(16911666217179/2968729, 200057852440887377496/5115120067)

Rank 10, proved: 10 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 13:20:26 UTC · history