curve #4178

y² + xy = x³ − 104802767424546495916x + 382567906903074864192504550800

a-invariants: [1, 0, 0, -104802767424546495916, 382567906903074864192504550800]
c-invariants: [5030532836378231803969, -330538671571802481916891279597153]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 2451861563535401352551788293184551769586091703999331590
discriminant (Δ): 10444537962810644097654209842839281010153616884364512680345600
Faltings height: 10.4510
naive height: 149.9094
primes of bad reduction: 2, 5, 13, 359, 6991, 7514824498054176890848361843367171577531611947
regulator: 48204073880141782.4472260602068196246649156400378942673354952246298112431
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:59:01 UTC
last updated: 2026-10-11 22:59:01 UTC

Witness: 17 independent points log in to add more points →

(18752694488, 2238711855562516)
(21187973464, 2770175542369556)
(5913133578564062/38809, 453698198113930383252048/7645373)
(-6383223220004312/654481, -365670735423731646107996/529475129)
(-1530870946186274/229441, 97361525705588433655470/109902239)
(-173594593045048/16129, 1052047502980306959468/2048383)
(3246408396450/29929, 3154590217705496061600/5177717)
(2437526089519/324, -823134173747791807/5832)
(12988664678197784/1692601, 382800160087082217502340/2202073901)
(18305267322239704/2181529, 987720064982452149022212/3222118333)
(-870988385819054/450241, -229697769260175201998030/302111711)
(3640110638, -222054021516734)
(-11949973886600/6561, -400316009841752415020/531441)
(-136169728052/121, -940296378034769204/1331)
(-152733408719/64, -402850714239995925/512)
(4707618479604/5041, -191243437389252965604/357911)
(2623987560800350084/45954841, -4186097433772416582558366260/311527867139)

Commentary

Mestre construction: sextuple [-1524, -1034, 94, 254, 780, 1430], T = 149; y^2 = r(x) with p(x-T)p(x+T) = g(x)^2 - r(x), p(x) = prod(x - a_i), on the locus sum a_i = 0, 12 p5 = 5 p2 p3. Found by a discriminant-first sweep + Nagao-sum rescoring + ratpoints search on the quartic.

last edited by Andrii Prokhorenkov at 2026-10-11 22:59:01 UTC · history