{
  "id": 6184,
  "curve_key": "3414920184023984272:-6312389242820558619226072024",
  "ainvs": [
    "0",
    "1",
    "1",
    "-71144170500499672",
    "7306006044364626753567322"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.02462351919,
  "faltings_height": 8.39802081714333,
  "conductor": "37096807484877365829693257492215827538671",
  "bad_primes": [
    "3",
    "29",
    "43",
    "30491",
    "92119",
    "3530432767011153628475732039"
  ],
  "discriminant": "-13050930684715903178723276951695078173127520730651",
  "regulator": "936845305762893512.5574581911267190010353783651708592457070676603062519",
  "points": [
    [
      "-1199135483/4",
      "10406748011093/8"
    ],
    [
      "-1521295009099/6241",
      "1571885367707517304/493039"
    ],
    [
      "-140544546526/5929",
      "1368010106552453594/456533"
    ],
    [
      "11178678377/64",
      "233677936584107/512"
    ],
    [
      "11308630289/64",
      "256948558622951/512"
    ],
    [
      "900259355261/45796",
      "23835291086885471977/9800344"
    ],
    [
      "532623855140/3481",
      "10303575169909874/205379"
    ],
    [
      "5259741984869/31684",
      "1496839454753445907/5639752"
    ],
    [
      "1540633994543/9409",
      "197572591835526145/912673"
    ],
    [
      "4141694772893/7921",
      "7495905766672849079/704969"
    ],
    [
      "16210888004285/196249",
      "122730025286248510376/86938307"
    ],
    [
      "16814815291616/105625",
      "4160525777135549086/34328125"
    ],
    [
      "51623498764853/337561",
      "9982203872099362448/196122941"
    ],
    [
      "196113154223864/1408969",
      "529158045215591616818/1672446203"
    ],
    [
      "451958434211816/3031081",
      "601415263832578710781/5277112021"
    ],
    [
      "96331567495457645/598193764",
      "2328238721561318846199977/14630623079912"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,9,16,33,43,56,61,83,104: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:41:15",
  "updated_at": "2026-10-30 14:41:15"
}