{
  "id": 6174,
  "curve_key": "123314318739330581200:-1401783456735796679884947320000",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-2569048307069387108",
    "1622434557263521518889966212"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 138.83059622678363,
  "faltings_height": 9.471680715930013,
  "conductor": "2713126118888294398313627472411615188672713800",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "19",
    "327421",
    "764809",
    "126574813501",
    "1072655525555227279"
  ],
  "discriminant": "-51983442175377342905801136098857098782665422953724000000",
  "regulator": "1038166723505131681954.32302077716042291783391941401419168053820873023929",
  "points": [
    [
      "539015088",
      "19856553758706"
    ],
    [
      "2160203278",
      "78443033391184"
    ],
    [
      "2317144098",
      "90059365712904"
    ],
    [
      "4356332386612",
      "9092461774151905450"
    ],
    [
      "-334118217987304/383161",
      "13415880900751524795102/237176659"
    ],
    [
      "-25368208261328/13689",
      "6980589274596676298/1601613"
    ],
    [
      "-21325033455644/17161",
      "120979850621639773022/2248091"
    ],
    [
      "-7669363734992/4489",
      "9627786623315340550/300763"
    ],
    [
      "-3489601746542/2209",
      "4329052538828696000/103823"
    ],
    [
      "-189844631687/729",
      "938571586201453400/19683"
    ],
    [
      "264964854712/361",
      "78870788237346350/6859"
    ],
    [
      "15153992812190782/18139081",
      "594686404016327112190400/77254345979"
    ],
    [
      "160597449920676082/30151081",
      "61736964830961835829099000/165559585771"
    ],
    [
      "204473406405697522/750705201",
      "631591407349942867453838800/20568571802199"
    ],
    [
      "209463182142946968/195077089",
      "27501801502091533088363850/2724641702063"
    ],
    [
      "346196175348408302/3517557481",
      "7723394440820274493718359800/208622816640629"
    ],
    [
      "7126339232138670172/8774256241",
      "6956736883002869111597895650/821893356350711"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,10,47,60,72,88,96,98,99,120: 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": "2026-10-30 14:40:03",
  "updated_at": "2026-10-30 14:40:03"
}