curve #2147

y² + xy = x³ − 54345303008894484183920562119085710x + 4874643038837841407702987096549823021442506758855172

a-invariants: [1, 0, 0, -54345303008894484183920562119085710, 4874643038837841407702987096549823021442506758855172]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 436526112602665716189006848580942310302569667141197757407610
discriminant (Δ): 7013806420802678404714712479061615866937699447140312900893222294104808977653879297346619153167424000000
Faltings height: 18.6986
naive height: 251.5556
primes of bad reduction: 2, 3, 5, 13, 43, 107, 113, 179, 193, 337, 647, 911, 1153, 1301, 1303, 4339, 5087, 5419, 7873, 30431, 66089, 84751
regulator: 6268910251678289.312784458177713245565922765138091991458795611108083129
submitted by: Steps Unbounded
submitted at: 2026-09-26 11:54:50 UTC
last updated: 2026-09-26 11:54:50 UTC

Witness: 16 independent points log in to add more points →

(45478878207780424, 49971440579345617184014498)
(1275205559445004, 69320586263369860597166818)
(-123704294250576776, 98510782241622571851178978)
(-72786210091833148676/361, 599348816631837996112421054422/6859)
(255925313240508838, 87913824793374674839678516)
(8393884563503825884, -24309651031321869715976770142)
(145396458058690714, -6835824561867189037938362)
(107060780950075684, 16838179394731332667120258)
(181676381578797163204/1681, -1118955629690076203339577143182/68921)
(167626135474583764, -21794413389730861289808062)
(170664161457724744, 23888397293679577418699938)
(96570325435725084, -22958607641096204241832542)
(16588786519327877344/25, -63857238895924499221835180038/125)
(705299254679193379156/4489, 4406233236889654893959050707214/300763)
(-51861404793245582564/529, 1170831298903329649876959269486/12167)
(-659243594953499729/4, 773755245533022244193841509/8)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=-688/2887. Found by a Mestre-Nagao sieve over t=a/b with |a|,b<=30000, screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-26 11:54:50 UTC · history