{
  "id": 6505,
  "curve_key": "5708252842922216568016:-607307559787071848033478115330240",
  "ainvs": [
    "0",
    "1",
    "0",
    "-118921934227546178500",
    "702902268232433253444380573984"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 150.9731762816791,
  "faltings_height": 10.604435894007773,
  "conductor": "13319113901119342563617347346081167126281944996568",
  "bad_primes": [
    "2",
    "3",
    "17",
    "151",
    "302322149",
    "396160157749",
    "1805084182240184448733271"
  ],
  "discriminant": "-105800869410754846603700885459678760198441160026722098539341568",
  "regulator": "909304895537398.58356067193965252913294086622144373323688229819218279604345",
  "points": [
    [
      "408038755580",
      "260554881377203872"
    ],
    [
      "489815234042",
      "-342722068716243906"
    ],
    [
      "1031328060857/16",
      "1033654000230093771/64"
    ],
    [
      "23536250624858/1369",
      "-97957890586064495178/50653"
    ],
    [
      "1021904380358/169",
      "-994489726707666522/2197"
    ],
    [
      "2293966366",
      "664959283070406"
    ],
    [
      "-111540715207030/12769",
      "1496147649747107789934/1442897"
    ],
    [
      "-850298777924/81",
      "-649785476136880576/729"
    ],
    [
      "-3672254562232/289",
      "-1979627350685739192/4913"
    ],
    [
      "-6046913362",
      "1095858813668490"
    ],
    [
      "-90455107569316/6889",
      "14343306374417520672/571787"
    ],
    [
      "86242116385070/24649",
      "2221899896693657056446/3869893"
    ],
    [
      "-12025374490",
      "627692669536578"
    ],
    [
      "112990641890",
      "37812754265788422"
    ],
    [
      "79403965228/9",
      "15753863414018888/27"
    ],
    [
      "39215742694697/256",
      "244979688916077528789/4096"
    ],
    [
      "-303397862734/529",
      "10682868644204433786/12167"
    ],
    [
      "677959272842/121",
      "613534443072301434/1331"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,16,37,52,60,113,135,170,174: 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 >= 18: the 18 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 04:54:50",
  "updated_at": "2026-11-04 04:54:50"
}