curve #3678

y² = x³ − 1154028987379000568180172x + 476874822756497046426048134610882736

a-invariants: [0, 0, 0, -1154028987379000568180172, 476874822756497046426048134610882736]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 3716012842930786646909643840
discriminant (Δ): 121683127663925100703696233445816670418082806365744147280371891343360000
Faltings height: 12.5753
naive height: 177.8295
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 37, 59, 109, 137, 157, 163, 199, 277, 449
regulator: 262728408.57869427728325803809422130764604041941356805625
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:39 UTC
last updated: 2026-09-29 19:31:39 UTC

Witness: 9 independent points log in to add more points →

(-1228249730203, 203411567839680585)
(465995200637, 200734707120348555)
(638827377752, 18847576795544820)
(761137413912, 198620355656933420)
(2812754631197, 4414094693347425435)
(-6639186955232764/7225, 535870489507141071035016/614125)
(-239486334726312/289, 4567079779170840855220/4913)
(141729115619057/196, 392781104293493253015/2744)
(75109015713764849/484, 20583903048061848895839465/10648)

Commentary

Specialization at t = 247/45 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:31:39 UTC · history