curve #4162

y² + xy + y = x³ − 14311842248200412599x + 22169060926184648345241903566

a-invariants: [1, 0, 1, -14311842248200412599, 22169060926184648345241903566]
c-invariants: [686968427913619804729, -19154068641253988812159434388333]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 781869987821636658363933019146211848405454531611164070
discriminant (Δ): -24699272915285502037716644074828832291128308653596672971300
Faltings height: 9.9526
naive height: 144.0601
primes of bad reduction: 2, 3, 5, 13, 35266397, 236451511265701, 240417872516149973451262990529
regulator: 34366661978483124.5014355097619062424133222199111811632176531104906832341
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 18:48:14 UTC
last updated: 2026-10-11 18:48:14 UTC

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

(-4204749791, -89480484328828)
(2180406313, -36462792086062)
(-1382079529096523/388129, 40447882224779851882066/241804367)
(6536163778447/3969, -13846435370658239894/250047)
(56945596196656/13225, -305614529470961588866/1520875)
(669051101617/529, -949600754867828324/12167)
(207736156259248/57121, -1842808774377294934853/13651919)
(-104266622324/25, -12062436965478191/125)
(898706472230599/292681, -13412095972847957306578/158340421)
(-112954054776431/47961, 2173296425611338914438/10503459)
(5377110968206846/9409, -394288668415162624555579/912673)
(7555671349, 587680602958958)
(3463880271, 118976435913526)
(1650720169, 55155532570628)
(9016083123887977/82369, -855601730234361334536836/23639903)
(203265823, 138810383799278)
(911076797701/361, 316112074424219444/6859)

Commentary

Mestre construction: sextuple [-345, -264, 33, 69, 195, 312], T = 271; 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 Nagao-sum sieve + ratpoints search on the quartic.

last edited by Andrii Prokhorenkov at 2026-10-11 18:48:15 UTC · history