{
  "id": 6014,
  "curve_key": "12698267178874245696:-44882217699670436030321512320",
  "ainvs": [
    "0",
    "0",
    "0",
    "-264547232893213452",
    "51947011226470412072131380"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 131.9639916463197,
  "faltings_height": 8.853217245685206,
  "conductor": "1684098819347899039402350571663824958629528",
  "bad_primes": [
    "2",
    "3",
    "19",
    "23",
    "769",
    "1087",
    "86719",
    "738386066630855250280816111"
  ],
  "discriminant": "19173141549629029109581888224457232819877923382189312",
  "regulator": "31804777057782.28719401379793791932843919746265278459656385668321264778",
  "points": [
    [
      "-457705278",
      "8783228955078"
    ],
    [
      "-281142006",
      "10202972809626"
    ],
    [
      "-236706006",
      "10065006494874"
    ],
    [
      "53701401",
      "6155917515927"
    ],
    [
      "54209214",
      "6145357188114"
    ],
    [
      "219047862",
      "2123405871822"
    ],
    [
      "229825074",
      "1812921455034"
    ],
    [
      "246002814",
      "1324799751186"
    ],
    [
      "319342914",
      "179678524986"
    ],
    [
      "335268804",
      "968774522706"
    ],
    [
      "385734438",
      "2701086381774"
    ],
    [
      "434220462",
      "4352738546322"
    ],
    [
      "584392890",
      "9845123474790"
    ],
    [
      "1049025057/4",
      "6224569918767/8"
    ],
    [
      "3070608033/4",
      "138849246809199/8"
    ],
    [
      "8364481761/25",
      "117851518570509/125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,9,17,19,34,44,55,57,62: 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 the sum of -a_p log(p)/p over 5 <= p <= 199; the sets scoring at least 28 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 04:53:57",
  "updated_at": "2026-10-28 04:53:57"
}