{
  "id": 6186,
  "curve_key": "14937980795499043776:-57962940995169046555798296960",
  "ainvs": [
    "0",
    "0",
    "0",
    "-311207933239563412",
    "67086737262927137217359140"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 132.459202737624,
  "faltings_height": 8.864277572957203,
  "conductor": "64327933600498682703332055234381017134072",
  "bad_primes": [
    "2",
    "7",
    "17",
    "37",
    "9305684629",
    "196251325291998358100491057"
  ],
  "discriminant": "-15273520381607007703035523252548575258654550773873408",
  "regulator": "4534766752812435032.987666254354042509655789825980511910555844522662621",
  "points": [
    [
      "1057811054",
      "30356881800434"
    ],
    [
      "-317497907065032/1100401",
      "13305242904728526395266/1154320649"
    ],
    [
      "-18175547919218/31329",
      "40129240027317143122/5545233"
    ],
    [
      "-4194765028318/908209",
      "7164734868207430547282/865523177"
    ],
    [
      "-3490950447938/6561",
      "4813568282214758962/531441"
    ],
    [
      "22746024656/121",
      "5193913924247902/1331"
    ],
    [
      "198106123689/625",
      "8401696170781613/15625"
    ],
    [
      "323364149466/625",
      "104312084900200514/15625"
    ],
    [
      "800938097566/2809",
      "184309876486317302/148877"
    ],
    [
      "21182681760526/112225",
      "145947384609261785774/37595375"
    ],
    [
      "29480213742946/93025",
      "15272695648596127006/28372625"
    ],
    [
      "61109018633561/131044",
      "229322393183854381117/47437928"
    ],
    [
      "383158583419846/3721",
      "7500003712022750839862/226981"
    ],
    [
      "780652876209694/55225",
      "21794838644024764317622/12977875"
    ],
    [
      "1556553421689320/6869641",
      "51574598767860667307830/18005329061"
    ],
    [
      "43870211678086328/154082569",
      "2389372346900321091161146/1912626928997"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,28,51,55,60,73,96,99,113: 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": "2026-10-30 14:41:29",
  "updated_at": "2026-10-30 14:41:29"
}