{
  "id": 6131,
  "curve_key": "4075288074919358659281:-262602200164258865821494141259929",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-84901834894153305402",
    "303937731692821368239156545401"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 149.29638626008267,
  "faltings_height": 10.303285068919969,
  "conductor": "6997011738463637227893675954536446387040433730",
  "bad_primes": [
    "2",
    "5",
    "29",
    "139",
    "6028103",
    "381358487",
    "75506740709120741075620403"
  ],
  "discriminant": "-739375954467144903311737691117621487498952399011971072000000",
  "regulator": "2604765951321901288.39070935738683406457185636274905302699058097415951314",
  "points": [
    [
      "-10588186211",
      "125925604536319"
    ],
    [
      "-10363567521",
      "265974873256089"
    ],
    [
      "-10265761999",
      "306033740025799"
    ],
    [
      "-674735471471/289",
      "3437117879077232727/4913"
    ],
    [
      "-107451028759/121",
      "819006325472085989/1331"
    ],
    [
      "137793247701/25",
      "7313501602675159/125"
    ],
    [
      "229504973345/49",
      "32589823978222561/343"
    ],
    [
      "243062393449/49",
      "23870318646876057/343"
    ],
    [
      "580671339521/121",
      "111497127455076269/1331"
    ],
    [
      "10159495396129/729",
      "26607529057334518117/19683"
    ],
    [
      "13526478604389/289",
      "48849649441389129087/4913"
    ],
    [
      "27123119140181/8281",
      "186105687545905011529/753571"
    ],
    [
      "30961647126305/5041",
      "42606091621568006177/357911"
    ],
    [
      "82992885732649/11449",
      "322717030610090095557/1225043"
    ],
    [
      "112002462777241/32041",
      "1280762785628366537261/5735339"
    ],
    [
      "197364462431041/16641",
      "2109068996806894868311/2146689"
    ],
    [
      "1256865571385409/253009",
      "8783922005129675645073/127263527"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,9,15,16,28,32,67,102,114: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 12:56:04",
  "updated_at": "2026-10-29 12:56:04"
}