{
  "id": 6523,
  "curve_key": "739563012618402081:-676130734272339671439255345",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-15407562762883377",
    "782558724074506051182945"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 123.55685736619174,
  "faltings_height": 8.2434580121112,
  "conductor": "27660077874604450688063675113319693118614",
  "bad_primes": [
    "2",
    "7",
    "6717853",
    "785621021",
    "374353390621395527956277"
  ],
  "discriminant": "-30466567624358428849883459592491111693978533756928",
  "regulator": "2295607762761.377377462620158677782821712920738482381294433467225492330",
  "points": [
    [
      "340558638295",
      "198740990547623124"
    ],
    [
      "83849479663/9",
      "-24278087318260516/27"
    ],
    [
      "1341872819639/289",
      "1553854323230113556/4913"
    ],
    [
      "33698372591/121",
      "-5661617000835188/1331"
    ],
    [
      "5981057563979/36100",
      "-11432166684188383897/6859000"
    ],
    [
      "1074853194743/12769",
      "-413378587757349740/1442897"
    ],
    [
      "60511912175/841",
      "-5257587176872444/24389"
    ],
    [
      "-40746250402205/820836",
      "-887756177849492530991/743677416"
    ],
    [
      "-248244182801/3025",
      "-203372848467893292/166375"
    ],
    [
      "4402465967/4",
      "290326593151951/8"
    ],
    [
      "320232441660679/2537649",
      "3721897258696627850132/4042474857"
    ],
    [
      "-18252127700441/483025",
      "384354230321553110508/335702375"
    ],
    [
      "-593060187343545/4800481",
      "9410866121376301210364/10517853871"
    ],
    [
      "95184905951/625",
      "21920661282675476/15625"
    ],
    [
      "416099671",
      "8149303051476"
    ],
    [
      "-5573111177828249/39262756",
      "81490604720841320055771/246020429096"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,17,77,95,113,116,151,165,185: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 16: the 16 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:56:57",
  "updated_at": "2026-11-04 04:56:57"
}