{
  "id": 6025,
  "curve_key": "3318327973643240016:-6145616254017992547469256640",
  "ainvs": [
    "0",
    "0",
    "0",
    "-69131832784234167",
    "7112981775483787670682010"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 127.97107306853445,
  "faltings_height": 8.551133281280922,
  "conductor": "2931503334443348555683450903039817245704936",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "439",
    "56774085483287",
    "17951545169341382075551"
  ],
  "discriminant": "-711511881582374334331425772412460198806903189453568",
  "regulator": "48273895717582.17794497195811123985283791625844218881583706976212716468",
  "points": [
    [
      "-297358111",
      "1173430795446"
    ],
    [
      "-254554651",
      "2866384684674"
    ],
    [
      "-41541823",
      "3148516230678"
    ],
    [
      "63591569",
      "1724510719386"
    ],
    [
      "128150063",
      "598556259756"
    ],
    [
      "135506321",
      "483056422758"
    ],
    [
      "169718499",
      "518329178276"
    ],
    [
      "192924227",
      "977941354572"
    ],
    [
      "221886069",
      "1642498972114"
    ],
    [
      "247486301",
      "2272048583238"
    ],
    [
      "261155117",
      "2621097082422"
    ],
    [
      "266079855",
      "2748898958260"
    ],
    [
      "374156705",
      "5798814738570"
    ],
    [
      "504667061",
      "10037803170798"
    ],
    [
      "-158702617214/729",
      "67743644687534998/19683"
    ],
    [
      "75393298739/529",
      "4792759510590324/12167"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,8,15,24,30,44,61,62,77: 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": "2026-10-28 11:29:11",
  "updated_at": "2026-10-28 11:29:11"
}