curve #3707

y² + xy = x³ − 7113973724879031416958740x + 7303503999847782915827520992153595792

a-invariants: [1, 0, 0, -7113973724879031416958740, 7303503999847782915827520992153595792]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/5ℤ
conductor (N): 177700463308819182300270
discriminant (Δ): -1567601625608890762147360135731913761034693551969414758912111158886400000
Faltings height: 12.9368
naive height: 183.2860
primes of bad reduction: 2, 3, 5, 13, 17, 19, 37, 41, 43, 331, 250091, 261241
regulator: 1214341.127162800473958601269035158014532898289053887
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:47:06 UTC
last updated: 2026-09-30 01:47:06 UTC

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

(809102756584, 1441264720297380268)
(1541932846264, 16349386579680028)
(-2013497389645466/841, 79450015774409150841482/24389)
(113850500896744/81, 207742533465272411852/729)
(1435301417006824/841, 8905772098481890034012/24389)
(3713893654348312/2401, 2547056226346889279980/117649)
(10489000725737896/9409, 794785738361285831394604/912673)

Commentary

Specialization at t = 3891105/152048 of the universal curve with a 5-torsion point, y^2 + (t+1)xy + ty = x^3 + tx^2 (as searched in arXiv:2003.00077, Sec. 12), 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 2026-09-30 01:47:06 UTC · history