{
  "id": 6102,
  "curve_key": "31988887267164681:-5051921844540382202028645",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-666435151399264",
    "5847131931123119102164"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 114.01249489348834,
  "faltings_height": 7.482763541966945,
  "conductor": "108919763789045844740103310230126931479554",
  "bad_primes": [
    "2",
    "7",
    "17",
    "23",
    "44884711987",
    "63329406057160191253919869"
  ],
  "discriminant": "4173587508868658678751278641398003760433550172",
  "regulator": "16789128430922159.44137558178765357984978568616462615563282218300268191",
  "points": [
    [
      "9826200",
      "15723014842"
    ],
    [
      "9904561",
      "14760715220"
    ],
    [
      "10509291",
      "2012572855"
    ],
    [
      "99965155",
      "968547893447"
    ],
    [
      "158129533",
      "1963205166992"
    ],
    [
      "672585658",
      "17429987086100"
    ],
    [
      "-166300332162/6241",
      "33755200548120610/493039"
    ],
    [
      "-17321264498/961",
      "3264189724228658/29791"
    ],
    [
      "-7187031624/289",
      "412299481424734/4913"
    ],
    [
      "-1417338033/49",
      "10425692205881/343"
    ],
    [
      "261973381/25",
      "471587956039/125"
    ],
    [
      "285970966/49",
      "15927292259724/343"
    ],
    [
      "12239611289/64",
      "1342230749276663/512"
    ],
    [
      "32590057551/841",
      "4767204756962066/24389"
    ],
    [
      "63154859542/81",
      "15862326323402420/729"
    ],
    [
      "1732370564401/54289",
      "1652648839107083236/12649337"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,9,15,26,28,46,55,67,73: 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.",
  "created_at": "2026-10-29 02:29:03",
  "updated_at": "2026-10-29 02:29:03"
}