curve #6183

y² + xy = x³ − 69283937711206345x + 7369712716702797558539801

a-invariants: [1, 0, 0, -69283937711206345, 7369712716702797558539801]
c-invariants: [3325629010137904561, -6367431792219660605785244905]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 29109434091169948293553656515843844039762
discriminant (Δ): -2177875148181136300520044234649013726059485978165248
Faltings height: 8.6065
naive height: 128.0420
primes of bad reduction: 2, 3, 29, 595207, 281071288182287246899399535819209
regulator: 714489547613882627.7294154584799142406449441296494732053126403111618102
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:41:08 UTC
last updated: 2026-10-30 14:41:08 UTC

Witness: 16 independent points log in to add more points →

(137806634, 662476613075)
(-98500696588726/398161, 769056184699938185381/251239591)
(-8484971345590/27889, 2497723768028482453/4657463)
(-1697661276106/7225, 2006712204679946627/614125)
(-121200481744/1225, 156105376646252599/42875)
(-1482783158/9, 102148114970801/27)
(116998923713/1024, 31854481872901327/32768)
(62886740218/1089, 67817413720772483/35937)
(149038334858/361, 48076933907749121/6859)
(205212773210/529, 75854051441531893/12167)
(227969309690/841, 71130475801460431/24389)
(10218350234321/53824, 12848023805545591883/12487168)
(4774481178554/26569, 3680263807068744737/4330747)
(6903318620954/7921, 17396155266375897595/704969)
(26811449679512/218089, 85820225460641418079/101847563)
(51515312095103834/134351281, 9494971074153054868333285/1557265698071)

Commentary

Mestre ten-root construction, roots 0,6,11,52,54,67,71,75,128,140: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:41:08 UTC · history