curve #6502

y² = x³ − 355134338176054624252x + 2529770195088837108478667023540

a-invariants: [0, 0, 0, -355134338176054624252, 2529770195088837108478667023540]
c-invariants: [17046448232450621964096, -2185721448556755261725568308338560]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 17781650931748464731843618509615062726355017661111990269304
discriminant (Δ): 101853296537055205984000246823075079296561541162849480262573312
Faltings height: 10.6887
naive height: 153.5707
primes of bad reduction: 2, 179, 18289, 89917, 7550875227399260691431851949878486369563079369
regulator: 1362623900856756772.029354389510767018000635580062221453325548828882729961697
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:28 UTC
last updated: 2026-11-04 04:54:28 UTC

Witness: 19 independent points log in to add more points →

(920374857820, -882788706267899030)
(-1517999323286/81, 1176204963351356126/729)
(1912442196734/25, -2570929267699455602/125)
(11009488660796/361, -30723401364246529138/6859)
(720480976786/25, -503594047732456534/125)
(2829954255258/361, -3271839682779740954/6859)
(88623750026554/9409, -130314193242982898674/912673)
(547314005478316/40401, 3676176938608650803798/8120601)
(15752485256513/1156, 18467234975775836801/39304)
(18002472500734/2025, 24990244733563588898/91125)
(-18327019570588/3481, -423577480467713871802/205379)
(42005402852763098/6806881, 13445808112974383599264174/17759152529)
(442806097681/64, -325375369949912057/512)
(211589291222665882/22934521, 21587059047022526252967478/109833421069)
(9645406286, 41287028073818)
(1506904390984186/156025, -1681563532602358367434/61629875)
(-3380587014, 1921379226651982)
(268857435935280754/33327529, -83832710209474485246826586/192399824917)
(77501353732198/16129, 1979909111045043844954/2048383)

Commentary

Mestre ten-root construction, roots 0,3,10,19,20,76,97,134,136,169: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 19: the 19 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:54:28 UTC · history