curve #6169

y² + xy = x³ − x² − 714621165290271150x + 167336948819766456647608400

a-invariants: [1, -1, 0, -714621165290271150, 167336948819766456647608400]
c-invariants: [34301815933933015209, -144579123625920046840835089173]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 82254192485037013616681222384133359549202218288370
discriminant (Δ): 11259776409276716793987492532164015588690291661494969300
Faltings height: 9.2706
naive height: 134.9452
primes of bad reduction: 2, 3, 5, 13, 37649, 508563435139, 3671754232125353251062853117451
regulator: 626953721580525584073.19230068903771176187624561108197589887931236168825618
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:25 UTC
last updated: 2026-10-30 14:39:25 UTC

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

(28784201080, 4881397117944400)
(36718095670, 7034037760791730)
(-15152176309194602/29822521, 3254256301875056041297384/162860787181)
(-688700247645850/966289, 16848435590989951213790/949862087)
(-660091906607513/822649, 11170769331411775371127/746142643)
(-177234846840992/192721, 578503324619372653396/84604519)
(-15780709938545/143641, 851289018800817011575/54439939)
(55556569399/49, 9789591733360132/343)
(56435626825/49, 10129008348890215/343)
(78154918999/25, 21097028752838708/125)
(90178528705/121, 9288067020375395/1331)
(6860570220394/529, 17932075426803074918/12167)
(16089980834680/67081, 54187705599734790880/17373979)
(22513442300512/29929, 38545163894971821476/5177717)
(63069846981022/69169, 301024104703960595408/18191447)
(83571470757286/121801, 6997958797475690296/42508549)
(1323172603979995/1885129, 8793603035155839409865/2588282117)
(7948092878466086/5958481, 579509529581124964567394/14544652121)

Commentary

Mestre ten-root construction, roots 0,6,14,17,21,25,74,92,106,121: 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 14:39:25 UTC · history