{
  "id": 6107,
  "curve_key": "2549982278576173529361:-128635297090978138220783767291641",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-53124630803670281862",
    "148883445720431780641639523349"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 147.87112008732953,
  "faltings_height": 10.096941145740983,
  "conductor": "178299557073213979216589428465303205829290",
  "bad_primes": [
    "2",
    "5",
    "11",
    "37",
    "43",
    "5621795918159",
    "181222586116878429887531"
  ],
  "discriminant": "19669932895234193814044623627919667570277216776912640000000",
  "regulator": "22249638960281.68053979994547425149946896653279060669855174219054182279",
  "points": [
    [
      "-8411169893",
      "25541480303121"
    ],
    [
      "-8372431743",
      "82334668516971"
    ],
    [
      "-1503146793",
      "474702003621771"
    ],
    [
      "-825552843",
      "438381514667571"
    ],
    [
      "2842860257",
      "144335430854771"
    ],
    [
      "3178055689",
      "110220430853403"
    ],
    [
      "3354791457",
      "91749673289771"
    ],
    [
      "3924835657",
      "28937344567771"
    ],
    [
      "3927238497",
      "28650155901171"
    ],
    [
      "4095948137",
      "2137942104091"
    ],
    [
      "4381108617",
      "15168851726811"
    ],
    [
      "16567144457",
      "1953437061311771"
    ],
    [
      "26034676707",
      "4051186688122521"
    ],
    [
      "36866636113/9",
      "51299080089817/27"
    ],
    [
      "45254912713/9",
      "2545953062979017/27"
    ],
    [
      "374535363837/16",
      "219213256945704369/64"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,14,19,58,69,79,88,92,106: 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": "2026-10-29 02:29:39",
  "updated_at": "2026-10-29 02:29:39"
}