{
  "id": 6070,
  "curve_key": "7051039600085232976:-19021200724221533864988460480",
  "ainvs": [
    "0",
    "1",
    "0",
    "-146896658335109020",
    "22015278567031592898700304"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 130.23070339104677,
  "faltings_height": 8.737356197770582,
  "conductor": "705887769261288306123559077880860758309448",
  "bad_primes": [
    "2",
    "3",
    "11",
    "3583",
    "85507965496198979",
    "8727268724481204701"
  ],
  "discriminant": "-6509500556737864265490714405231687358057248903383808",
  "regulator": "676798554777838267.4720025937223963719063566441319079710603303588113357",
  "points": [
    [
      "356866018",
      "3878264790390"
    ],
    [
      "415492643",
      "5719179269760"
    ],
    [
      "-345337760332602883/1604322916",
      "424608007244653165150934541/64259550077464"
    ],
    [
      "-440966640765664/1442401",
      "10727948639848035118644/1732323601"
    ],
    [
      "-4499788072654/17161",
      "14656628401128980046/2248091"
    ],
    [
      "-65209754554/1681",
      "362444215062569574/68921"
    ],
    [
      "34228626854/289",
      "12310484038203978/4913"
    ],
    [
      "43064266118/169",
      "2334687994463358/2197"
    ],
    [
      "90746063093/4",
      "27332547584877657/8"
    ],
    [
      "130850920490/529",
      "10977003852911934/12167"
    ],
    [
      "614790055328/2809",
      "87924469817062692/148877"
    ],
    [
      "9856827380162/61009",
      "23823648279705423930/15069223"
    ],
    [
      "26845625149853/98596",
      "45962009003408534685/30959144"
    ],
    [
      "653112407748923/2968729",
      "3008883641506111349748/5115120067"
    ],
    [
      "738014286682358/6713281",
      "46657106509132346058150/17394111071"
    ],
    [
      "362159905064505134/1594005625",
      "38627038687123766329203402/63640674578125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,15,35,42,44,55,75,81,109: 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": "2026-10-28 21:14:14",
  "updated_at": "2026-10-28 21:14:14"
}