{
  "id": 6068,
  "curve_key": "4367081076796423511409:-288669667049777702702368969642377",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-90980855766592156488",
    "334108410959987925402723235467"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 149.4856716071971,
  "faltings_height": 10.184001476987936,
  "conductor": "57135593756391946893120530343475250404125239830",
  "bad_primes": [
    "2",
    "5",
    "7",
    "11",
    "31",
    "293",
    "76862187676199",
    "106285486374011684312128487"
  ],
  "discriminant": "-25369881019170372075833759456482477353310990825640948326400",
  "regulator": "167826353034492610530.277556700124212037412471497187048251433188681255359",
  "points": [
    [
      "-3619860877",
      "784867398713483"
    ],
    [
      "-10364502244571/5625",
      "296963239642294851067/421875"
    ],
    [
      "19562980575013/3481",
      "3557107364367974507/205379"
    ],
    [
      "123819409370077/29584",
      "830468493037170173719/5088448"
    ],
    [
      "39652614444437/4489",
      "140967216199995448579/300763"
    ],
    [
      "1059648043495653/196249",
      "1448054615810995565259/86938307"
    ],
    [
      "4638754457254621/848241",
      "8269152424364201032129/781229961"
    ],
    [
      "84838095072161453/16313521",
      "2644697500874108068512607/65890311319"
    ],
    [
      "158203038360043637/19123129",
      "32131117672739190492885301/83625443117"
    ],
    [
      "169253655488210287/59644729",
      "144776187379784714079897051/460636242067"
    ],
    [
      "362550067338225553/69772609",
      "23701010295547676244032605/582810602977"
    ],
    [
      "471295738007895397/6661561",
      "320748534157242270105102565/17193488941"
    ],
    [
      "629257476171379053/92179201",
      "156191006424683205297851993/885012508801"
    ],
    [
      "908250285555910977/230827249",
      "674932437787500307262913701/3506958394057"
    ],
    [
      "8114696529793821003/1457254276",
      "682523873804229440555897947/55629224732024"
    ],
    [
      "2707740550120852917/463670089",
      "441386888713843107260594941/9984208026437"
    ],
    [
      "10316307255344053517/1903838689",
      "1220739944610648814388457161/83070193517137"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,20,21,26,35,60,75,90,106,131: 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": "2026-10-28 21:13:50",
  "updated_at": "2026-10-28 21:13:50"
}