curve #6490

y² + xy = x³ − x² − 28142489439385550143524x + 8405803115499779884518863965587724

a-invariants: [1, -1, 0, -28142489439385550143524, 8405803115499779884518863965587724]
c-invariants: [1350839493090506406889161, -7262613891785731042505391187436792325]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 196347104202630277828224678520976443046562484852749066157604334
discriminant (Δ): -29097565253392985165480954342019770950545551185885388245172550942258348
Faltings height: 12.2026
naive height: 169.7516
primes of bad reduction: 2, 3, 7, 127, 14353421, 7708862556316319877716933, 110893095984415313988881719
regulator: 4752201857280451808.56271155172708068957694213552375844824990955177053453695541
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:04 UTC
last updated: 2026-11-04 04:53:04 UTC

Witness: 20 independent points log in to add more points →

(5876731876445/16, -13966044415520067583/64)
(23357925907135817, -3569861903138058639964912)
(1144115221051060/1089, -38344740227667867353494/35937)
(16079830850094269/10000, -2029968700303856031546847/1000000)
(-2893958874012691/21904, 321211581193622610278705/3241792)
(-595391014632109/2500, 5000404592164329193789/125000)
(-2500184025650053/27556, -462259312545501688618465/4574296)
(1276413502936235/3364, 44643730147036337744981/195112)
(29355083690408789/35344, 4963365687244094194119145/6644672)
(709583296672455548/2493241, -602847485083792325462354902/3936827539)
(530065979974305/1849, -12291032535386656956379/79507)
(43063377763235597/85849, -8731883541376489633576445/25153757)
(483026826040835/2116, -11466126083657369136913/97336)
(26796656668095338/332929, -15679537647748114489064710/192100033)
(6495298574990903/20164, -518288239772822646365671/2863288)
(4764362545969046/10609, -321081253231758926958934/1092727)
(2388697855670964/1681, -116101672443446815659542/68921)
(-141717295325001958/727609, -50031999534095396981192590/620650477)
(222508141322204/289, -3270512945174438893490/4913)
(362476875717, -214080660612631105)

Commentary

Mestre ten-root construction, roots 0,4,99,100,148,150,166,174,188,197: 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 >= 20: the 20 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 04:53:04 UTC · history