curve #6170

y² + xy + y = x³ − x² − 1317837124555501486862x + 17527825124158781768840206873885

a-invariants: [1, -1, 1, -1317837124555501486862, 17527825124158781768840206873885]
c-invariants: [63256181978664071369361, -15144040906988534629373950417874745]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 174117693036610424661072980305410965823814139317254
discriminant (Δ): 13754516528009961126928258706015332744857844843096473684280983552
Faltings height: 11.0630
naive height: 157.5044
primes of bad reduction: 2, 3, 7, 11, 139, 227, 281, 2267, 6250016683150324800955592393624539069
regulator: 3512130842499827869410.3526012722933445876108697295973084380057956384840232
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:33 UTC
last updated: 2026-10-30 14:39:33 UTC

Witness: 18 independent points log in to add more points →

(-20841392259040239431/1388829289, 301460726601944158676471158073/51757501113163)
(-2911952346908634623/100620961, 5658409360746517134270420757/1009328859791)
(-430256846813219855/12538681, 209875721769642908164374839/44399469421)
(-198491323162019/4761, 22319221690233865139/328509)
(32989816717/1369, 211872895993371842627/50653)
(3808719957373/121, 3579791609169655837/1331)
(7546750581665/169, 15177785908165319503/2197)
(15379386654497/961, 21820404224928407197/29791)
(799543110374437/29584, 6540221611990968923657/5088448)
(439400878619545/3481, 8863348279870931373961/205379)
(850301936235823/1681, 24732481534134793637989/68921)
(6041119353683749/4225, 469393671833479140289963/274625)
(99470876978072065/5822569, 211950714954298778513903/14049858997)
(351720525761647681/52548001, 1143203484548324900354989523/380920459249)
(13418585900892702889/14799409, 49115282128428368994685895861/56933326423)
(21542859625599730969/722480641, 42296939749245462325255123885/19419557149439)
(80980578438616118273/796876441, 687138120660376936111241601943/22495025052989)
(2250969825953868407413/9318427024, 105650071282939021657033623368141/899526397480768)

Commentary

Mestre ten-root construction, roots 0,1,65,67,79,88,98,118,129,139: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:39:33 UTC · history