{
  "id": 6469,
  "curve_key": "441420818360614972041:-9189391093731225247503476434725",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-9196267049179478584",
    "10635869323747244132090559964"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 142.60959099297207,
  "faltings_height": 9.745467844885786,
  "conductor": "154914169134480285882262406943796131306286378249214",
  "bad_primes": [
    "2",
    "7",
    "23",
    "53",
    "5711",
    "6599071",
    "321677632097",
    "748761621964991059060147"
  ],
  "discriminant": "906811467184020911691274751257668898467467582601972540532",
  "regulator": "84688185526665918.857788020792408579631969295032586313664862747553442279520",
  "points": [
    [
      "14876754622/81",
      "68978300091257330/729"
    ],
    [
      "159806093669",
      "-63872351278537997"
    ],
    [
      "-21149460602/9",
      "3748042502474080/27"
    ],
    [
      "888068308718/289",
      "-524415728740220640/4913"
    ],
    [
      "5768569390335/2809",
      "-3018413870105463661/148877"
    ],
    [
      "27677016788013/21904",
      "-104205060694316974895/3241792"
    ],
    [
      "8670998552682/5929",
      "-8097278538095059354/456533"
    ],
    [
      "31924545677076/9025",
      "128225921393209203614/857375"
    ],
    [
      "58030405793491/12321",
      "366464627115206370554/1367631"
    ],
    [
      "-3465323577",
      "29846075717690"
    ],
    [
      "-3161066477",
      "90109436890215"
    ],
    [
      "1219595417",
      "35130376965580"
    ],
    [
      "1886159245",
      "661708977987"
    ],
    [
      "10806659751/4",
      "593828052014639/8"
    ],
    [
      "4735078220",
      "270655815579318"
    ],
    [
      "12734930115",
      "1399584006422407"
    ],
    [
      "16132854561",
      "2015222861815339"
    ],
    [
      "339390616187/169",
      "35884465153894830/2197"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,24,34,53,74,77,95,111,129: 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.",
  "created_at": "2026-11-04 02:10:24",
  "updated_at": "2026-11-04 02:10:24"
}