{
  "id": 6124,
  "curve_key": "81074242752951944348881:-20713780488401708587321618003269529",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1689046724019832173935",
    "23974282972546408823139108896697"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 158.24895679483188,
  "faltings_height": 11.16258067331877,
  "conductor": "19856314773807030314011608433490767063472777910",
  "bad_primes": [
    "2",
    "3",
    "5",
    "31",
    "6309830796427981",
    "3383747803975121801713397527"
  ],
  "discriminant": "60094304286211659254645369557334274584645061565765012800000000000",
  "regulator": "209775667167923164.122586334626158046320205275825644278125173385157476296",
  "points": [
    [
      "-21714821306",
      "7100177464191613"
    ],
    [
      "12709641934",
      "2135445795436333"
    ],
    [
      "15075091414",
      "1392001828722013"
    ],
    [
      "15316127884",
      "1302890042350183"
    ],
    [
      "15822104134",
      "1100400747631933"
    ],
    [
      "131784563494",
      "45717644791806493"
    ],
    [
      "-12262119422876/361",
      "44534155097857120447/6859"
    ],
    [
      "-6573451578606/841",
      "147747409331081268237/24389"
    ],
    [
      "-32175813794/9",
      "147803985618473191/27"
    ],
    [
      "63769487356/9",
      "94931947391873791/27"
    ],
    [
      "517169642854/81",
      "2673581871719335157/729"
    ],
    [
      "14256509682726/289",
      "38276643561401498829/4913"
    ],
    [
      "28976225300266/1849",
      "92495729429934588031/79507"
    ],
    [
      "42200796027646/169",
      "270620525558066602801/2197"
    ],
    [
      "44114442150886/529",
      "261816476081400184811/12167"
    ],
    [
      "1062248560842871/1444",
      "34567921593075924197951/54872"
    ],
    [
      "3956143448842204/36481",
      "232784685956466632816393/6967871"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,18,23,42,71,73,84,85,113,137: 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 03:14:30",
  "updated_at": "2026-10-29 03:14:30"
}