{
  "id": 6495,
  "curve_key": "4601110054731579949881841:-9394705686211801538080910073305974345",
  "ainvs": [
    "1",
    "0",
    "0",
    "-95856459473574582289205",
    "10873501951626041519674551740024481"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 170.36501946745818,
  "faltings_height": 12.1346802001665,
  "conductor": "1527997307074255987966942096130361721461753649439001966",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "2851",
    "25849",
    "21712742543531194873",
    "2066926182404924270286859"
  ],
  "discriminant": "5292817332086209334734600129167610796935409470813874421081496062124032",
  "regulator": "58658407570421899.81579326650949625365950530991153560160103614838053916907909",
  "points": [
    [
      "214053308795965/784",
      "1560686602366144318183/21952"
    ],
    [
      "17243810485708",
      "-71594615598627890441"
    ],
    [
      "524884586388031/2500",
      "215203793040007798529/125000"
    ],
    [
      "61200528411586/289",
      "-41396953396601660857/4913"
    ],
    [
      "51000923720398/361",
      "-84277142998374485963/6859"
    ],
    [
      "-128463230094506/529",
      "-1713368280550748322103/12167"
    ],
    [
      "2962040264527969/2304",
      "156887685081088639683097/110592"
    ],
    [
      "34384963312751206/22201",
      "6256885819402408028869379/3307949"
    ],
    [
      "54677556892710130/26569",
      "12647922812157497566527853/4330747"
    ],
    [
      "357612851920",
      "149425447114235599"
    ],
    [
      "4056660168229/16",
      "3427650814532782039/64"
    ],
    [
      "276690053023218994/543169",
      "122882621445724874702405303/400315553"
    ],
    [
      "136653106105506/289",
      "1311593768308346643975/4913"
    ],
    [
      "3092546287765218/10201",
      "101347492700677240178043/1030301"
    ],
    [
      "-47868402547482866/305809",
      "25107751712700516377287771/169112377"
    ],
    [
      "214484602726",
      "-13447813478072177"
    ],
    [
      "324514801582887430/5004169",
      "786000067038592195003132043/11194326053"
    ],
    [
      "311225437318",
      "105764798077543339"
    ],
    [
      "302259061507162/289",
      "5045502038728841761079/4913"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,28,50,51,57,69,144,149,163,195: 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 04:53:39",
  "updated_at": "2026-11-04 04:53:39"
}