{
  "id": 6477,
  "curve_key": "156432038500857:-969465928089902914701",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-3259000802101",
    "1122067861150551121"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 98.05092831843191,
  "faltings_height": 6.222223110908737,
  "conductor": "1579774424461700958540633262007143658",
  "bad_primes": [
    "2",
    "23",
    "47819",
    "93419",
    "123791",
    "62102984510672533573"
  ],
  "discriminant": "1671401341080479614135989991203557990164",
  "regulator": "18526154427834.016986490889657087244233609882438558282657885606304697",
  "points": [
    [
      "43840179/25",
      "111668818348/125"
    ],
    [
      "63037405379/4",
      "-15827022538298343/8"
    ],
    [
      "1511174611/900",
      "16704354365219/27000"
    ],
    [
      "2794412764/961",
      "-120069078549789/29791"
    ],
    [
      "-21503450381/16900",
      "3936918228893493/2197000"
    ],
    [
      "-235600301/121",
      "391036858075/1331"
    ],
    [
      "7367085739/625",
      "624996808310588/15625"
    ],
    [
      "4087856958/169",
      "260617540942097/2197"
    ],
    [
      "319288400047/2401",
      "180391453965267189/117649"
    ],
    [
      "-629426",
      "1710286421"
    ],
    [
      "-323016",
      "1463402401"
    ],
    [
      "15",
      "1059253964"
    ],
    [
      "-5574500/9",
      "46011738703/27"
    ],
    [
      "181092536/529",
      "2622489364967/12167"
    ],
    [
      "913180723/529",
      "9725196588964/12167"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,15,45,62,65,66,75,78,98,116: 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 15, proved: 15 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:11:24",
  "updated_at": "2026-11-04 02:11:24"
}