curve #6123

y² = x³ + x² − 116172413289044375580x + 497501433975962547175446953424

a-invariants: [0, 1, 0, -116172413289044375580, 497501433975962547175446953424]
c-invariants: [5576275837874130027856, -429841238988689295786830947925440]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 7767535589308845848132270205771462688836499272
discriminant (Δ): -6579871306915912852402074324636225017391611245834473672284928
Faltings height: 10.4400
naive height: 150.2819
primes of bad reduction: 2, 3, 7, 13, 31, 16389695458611531389013247172084077515649
regulator: 14146575926716997303.9669080931398975669680761443256454104557144366652353
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:14:23 UTC
last updated: 2026-10-29 03:14:23 UTC

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

(132816330999, 48249058008107502)
(-134195912668897182/16024009, 60276625626508970085682002/64144108027)
(-517147773254514/43681, 4217621861798319138606/9129329)
(-336069704268801/34969, 5573049322301087629842/6539203)
(-4322999627739/5476, 310920941681574706161/405224)
(-1505063254626/121, 178869001412728458/1331)
(3222311341446/529, 1532824286231436534/12167)
(8644534667630/1369, 6348456344990076954/50653)
(15584648941989/3364, 47284898269394333457/195112)
(15621789680442/361, 59993687545512121674/6859)
(78180799894950/5329, 543344221460702850006/389017)
(93774572310939/19321, 588326405487254578452/2685619)
(208057568970267/19321, 1889906955629467912956/2685619)
(2865303805156146/410881, 42869571983239264978806/263374721)
(4495127821774704/201601, 271443435237693166747572/90518849)
(21560566217646726/3530641, 833919242405237273649546/6634074439)
(240163182251355600/13756681, 99334676387901587417680428/51023529829)

Commentary

Mestre ten-root construction, roots 0,8,61,72,92,114,115,117,130,139: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 03:14:23 UTC · history