{
  "id": 6494,
  "curve_key": "112407054136135359907761:-37955182802795369699592028545886185",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-2341813627836153331412",
    "43929609726043057225931293594295"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 159.24342826512807,
  "faltings_height": 11.120473380754412,
  "conductor": "275993669292054687436741062820989156987142998942730458",
  "bad_primes": [
    "2",
    "3",
    "167",
    "10600957",
    "8660940822751677874735696528448967274511599"
  ],
  "discriminant": "-11744155107108712633724189089819326917844926506042596750627176448",
  "regulator": "117240687046943376.4442596274355971032016844565085835990997415473934405642927",
  "points": [
    [
      "3203111169537/121",
      "-929642101086481885/1331"
    ],
    [
      "1340195381925",
      "-1550504484546381971"
    ],
    [
      "86100285853701/3481",
      "219142733553900070943/205379"
    ],
    [
      "2615805162339/100",
      "755505988416635293/1000"
    ],
    [
      "238029209287533/6889",
      "1180828690832860137455/571787"
    ],
    [
      "117055048942917/2401",
      "794769716817420480445/117649"
    ],
    [
      "31381270296621/361",
      "152948427865874864623/6859"
    ],
    [
      "16253795518957/169",
      "58471556759498204529/2197"
    ],
    [
      "679236030309069/4489",
      "16890624044366228987071/300763"
    ],
    [
      "115271159377941/5041",
      "-547118500435548348469/357911"
    ],
    [
      "-2480920210011/49",
      "-1961560547031382037/343"
    ],
    [
      "-7575457608387/1849",
      "-581300846735588130281/79507"
    ],
    [
      "13058321079925/529",
      "-13125679451459246085/12167"
    ],
    [
      "906879861153/361",
      "-42316490578498822517/6859"
    ],
    [
      "5804598909165/169",
      "-4402128522359394575/2197"
    ],
    [
      "5754945252032205/223729",
      "-89258120783641877211667/105823817"
    ],
    [
      "3338659042173/121",
      "-753620922519581641/1331"
    ],
    [
      "1059634744793317/83521",
      "97334528431199143090125/24137569"
    ],
    [
      "-8889212627283/169",
      "-10206901712251439567/2197"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,33,59,60,83,98,114,117,134,186: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 19: the 19 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:53:32",
  "updated_at": "2026-11-04 04:53:32"
}