curve #4177

y² + xy + y = x³ − 26226342675618574099x + 44493515720160243940176209922

a-invariants: [1, 0, 1, -26226342675618574099, 44493515720160243940176209922]
c-invariants: [1258864448429691556729, -38442397584106747436956782707917]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 990433349479246683456268348048692288078629191071270710
discriminant (Δ): 299279245212143970339980606729873348688719382666005870440700
Faltings height: 10.1366
naive height: 145.7535
primes of bad reduction: 2, 5, 11, 41, 67, 8707, 378347134263144107, 994981297502993806984589287
regulator: 30819873128172819.8799654030780724146344001361794390765454160960733710017
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:58:56 UTC
last updated: 2026-10-11 22:58:56 UTC

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

(4144962871, 83662340450254)
(4998326841, -195655861648786)
(1847165842653/361, 1443063624711797886/6859)
(449277978350061/312481, -17255213929143676452784/174676879)
(1806714132957816/72361, -75274792687022960608674/19465109)
(-19683442502426/5329, -117332796151589979432/389017)
(1419245171244209/134689, -47877168019603983942168/49430863)
(-5660538457, -107586605678238)
(3142280721308371/320356, -155023028560612128894151/181321496)
(701705236452001/1666681, 393977404782721055052954/2151685171)
(1174612756, 123726329966289)
(75018453305689/81, 649748530288799565400/729)
(-13349991530/9, 7643078625652456/27)
(544935598, 174251468515504)
(-4455113084149902489/7238436241, 151353992964082012973950853506/615838916948039)
(45925937095/81, 125856940910347214/729)
(111603909314059759/55756089, -1739118607466305901417498/416330716563)

Commentary

Mestre construction: sextuple [-803, -792, 315, 365, 427, 488], T = 517; 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 22:58:56 UTC · history