curve #6099

y² + xy = x³ − 85813162071400397985x + 304017541558957655838506486601

a-invariants: [1, 0, 0, -85813162071400397985, 304017541558957655838506486601]
c-invariants: [4119031779427219103281, -262671155913117962313610433078185]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 10782026790949638310422004913289526079203206498
discriminant (Δ): 514524622327718493667291975885123816321503115017152951887872
Faltings height: 10.2885
naive height: 149.3097
primes of bad reduction: 2, 3, 13, 53, 61, 42756298392973257791929401577044128574727
regulator: 29275317706333984701.4819186879684299487996173943824920253225695184570549
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:28:40 UTC
last updated: 2026-10-29 02:28:40 UTC

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

(-5000031294, 779799711022851)
(10386290466378, 33472677714834589323)
(-188751631147710/63001, 11558009055184035007929/15813251)
(-175619170754882/16641, 397354836786386889251/2146689)
(-38875955745894/3721, 55669756569240992151/226981)
(-276307031350/169, 1457240066075579447/2197)
(-171026903430/49, 256910557926132573/343)
(1252919446590/289, 570217928750163123/4913)
(12206915492298/3481, 44152264875959473137/205379)
(60984772054554/21025, 859648436808685384827/3048625)
(72525034986762/3721, 557640251889954488679/226981)
(127295942203878/1849, 1423834371273297355893/79507)
(945656097524769/430336, 100228687532242729830375/282300416)
(2940964668504978/383161, 74076691711807561469337/237176659)
(3110448390189834/452929, 59870839983610200364587/304821217)
(154173901186917867/15194404, 40946977852323457689209703/59227786792)
(119743597702589430/5822569, 37796600139189460103525583/14049858997)

Commentary

Mestre ten-root construction, roots 0,8,21,30,31,33,83,90,112,124: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 02:28:40 UTC · history