{
  "id": 6099,
  "curve_key": "4119031779427219103281:-262671155913117962313610433078185",
  "ainvs": [
    "1",
    "0",
    "0",
    "-85813162071400397985",
    "304017541558957655838506486601"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 149.309715250875,
  "faltings_height": 10.28846745044791,
  "conductor": "10782026790949638310422004913289526079203206498",
  "bad_primes": [
    "2",
    "3",
    "13",
    "53",
    "61",
    "42756298392973257791929401577044128574727"
  ],
  "discriminant": "514524622327718493667291975885123816321503115017152951887872",
  "regulator": "29275317706333984701.4819186879684299487996173943824920253225695184570549",
  "points": [
    [
      "-5000031294",
      "779799711022851"
    ],
    [
      "10386290466378",
      "33472677714834589323"
    ],
    [
      "-188751631147710/63001",
      "11558009055184035007929/15813251"
    ],
    [
      "-175619170754882/16641",
      "397354836786386889251/2146689"
    ],
    [
      "-38875955745894/3721",
      "55669756569240992151/226981"
    ],
    [
      "-276307031350/169",
      "1457240066075579447/2197"
    ],
    [
      "-171026903430/49",
      "256910557926132573/343"
    ],
    [
      "1252919446590/289",
      "570217928750163123/4913"
    ],
    [
      "12206915492298/3481",
      "44152264875959473137/205379"
    ],
    [
      "60984772054554/21025",
      "859648436808685384827/3048625"
    ],
    [
      "72525034986762/3721",
      "557640251889954488679/226981"
    ],
    [
      "127295942203878/1849",
      "1423834371273297355893/79507"
    ],
    [
      "945656097524769/430336",
      "100228687532242729830375/282300416"
    ],
    [
      "2940964668504978/383161",
      "74076691711807561469337/237176659"
    ],
    [
      "3110448390189834/452929",
      "59870839983610200364587/304821217"
    ],
    [
      "154173901186917867/15194404",
      "40946977852323457689209703/59227786792"
    ],
    [
      "119743597702589430/5822569",
      "37796600139189460103525583/14049858997"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,8,21,30,31,33,83,90,112,124: 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 02:28:40",
  "updated_at": "2026-10-29 02:28:40"
}