{
  "id": 6479,
  "curve_key": "642754902888361:-16465086559424071531925",
  "ainvs": [
    "1",
    "1",
    "0",
    "-13390727143507",
    "19056807568011921145"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 102.31105825173351,
  "faltings_height": 6.392220861812904,
  "conductor": "3170461919211389746836541940684033082",
  "bad_primes": [
    "2",
    "3",
    "13",
    "95737",
    "3251761",
    "50991937109",
    "2560518957463"
  ],
  "discriminant": "-3214848386080349203292253527853609545148",
  "regulator": "1743764451032.6421777905399317287370045955760648089935680678058177329",
  "points": [
    [
      "537830089/25",
      "12301904634152/125"
    ],
    [
      "25548703",
      "-127893601148"
    ],
    [
      "157165921/9",
      "1929945117973/27"
    ],
    [
      "2474349",
      "-1036826176"
    ],
    [
      "1004393536/529",
      "-8413482405811/12167"
    ],
    [
      "-6266984/25",
      "591595039649/125"
    ],
    [
      "-6535680428/3481",
      "1259214421200503/205379"
    ],
    [
      "-150582207/64",
      "-3136334674541/512"
    ],
    [
      "-10679165015/3721",
      "-1320239934301325/226981"
    ],
    [
      "815538",
      "2945534815"
    ],
    [
      "1592353",
      "1330217527"
    ],
    [
      "2219616",
      "518421685"
    ],
    [
      "2905756",
      "2162142439"
    ],
    [
      "59302936/25",
      "99867557351/125"
    ],
    [
      "209889640/49",
      "2176473176485/343"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,6,21,23,35,59,84,122,131: 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 15, proved: 15 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:11:38",
  "updated_at": "2026-11-04 02:11:38"
}