curve #6101

y² + xy + y = x³ − x² − 56218417291767046322x + 160932505855565946370965000625

a-invariants: [1, -1, 1, -56218417291767046322, 160932505855565946370965000625]
c-invariants: [2698484030004818223441, -139045685047065799529492078534745]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 15042348617841095595823355422402529435108652138
discriminant (Δ): 182964014843935538155341625068220986965911853390976374341632
Faltings height: 10.1927
naive height: 148.0409
primes of bad reduction: 2, 3, 11, 313, 6271, 55441, 2174741, 321018989709898828986366637
regulator: 134341422699397113058.554611561303551637181914443484032584412709827115853
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:28:56 UTC
last updated: 2026-10-29 02:28:56 UTC

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

(5705390131777, 13627856573091991727)
(-35371040878452006726575/6877668845961, 10107663185076602949879305630608525/18036899756266947291)
(-85721404382735639/17131321, 39919571680846701156712189/70906537619)
(-229586981591383/26569, 167326587586137487349/4330747)
(-112610142801623/17161, 1117922488570261792405/2248091)
(362534814121/25, 193454135751570691/125)
(167422029177521/9025, 2011094408790617724721/857375)
(181311911513801/10201, 2251787401739011648451/1030301)
(300555135386593/96721, 3833623671194386077337/30080231)
(3209475894030121/687241, 8840756552466835108523/569722789)
(8087708820862921/133225, 722034464226775585310299/48627125)
(21493884922758529/1408969, 2825093389446061808336293/1672446203)
(22716088720035295/6355441, 1204866906252662922781385/16022066761)
(103048409116308193/21986721, 2059554222045356891560991/103095734769)
(112918152978254401/28185481, 420831374552072597636923/149636718629)
(264407833393479881/53333809, 24850692167941189521936545/389496807127)
(271773475686033745/26697889, 110659178805647723442150977/137947992463)

Commentary

Mestre ten-root construction, roots 0,3,30,44,85,98,101,106,118,127: 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 02:28:56 UTC · history