{
  "id": 3009,
  "curve_key": "31189362083877005223264:-5508107958770570755736866577872512",
  "ainvs": [
    "0",
    "0",
    "0",
    "-649778376747437608818",
    "6375124952280753189510262242908"
  ],
  "rank_lower_bound": 13,
  "torsion": [
    2
  ],
  "naive_height": 155.38309209210183,
  "faltings_height": 10.591291477274146,
  "conductor": "1005791021107376101017459673931692686720",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "23",
    "67",
    "71",
    "73",
    "137",
    "151",
    "401",
    "619",
    "1583",
    "2789",
    "4513",
    "22303"
  ],
  "discriminant": "589806933121145449993043501676982102463659315457163169171200",
  "regulator": "341975471789.8617545759339195814407451384181045658293982040004978",
  "points": [
    [
      "58199682499",
      "12872154885340815"
    ],
    [
      "22019376748",
      "1656372511724994"
    ],
    [
      "14455983388",
      "53714508667686"
    ],
    [
      "65767704577/4",
      "-2954338331079537/8"
    ],
    [
      "14361000028",
      "-73798447287366"
    ],
    [
      "57888581689/4",
      "-402173765593443/8"
    ],
    [
      "1064749219",
      "2384214971351265"
    ],
    [
      "13579625186",
      "-235680916199704"
    ],
    [
      "-4218467869/25",
      "-318314970685386529/125"
    ],
    [
      "-4546387891364/289",
      "17511165320346426330/4913"
    ],
    [
      "3868085294579/289",
      "-1353922157643031127/4913"
    ],
    [
      "89134685321266/1849",
      "-741932914883664155760/79507"
    ],
    [
      "6878345735617/4",
      "-18037591725212102673/8"
    ]
  ],
  "submitter": "Steps Unbounded",
  "history": [],
  "commentary": "Rank 13, torsion Z/2Z. Fibre (a,b,c,u) = (45,130,170,157) of Kihara's Z/2Z construction (six points from the quartic y^2 = A t^4 + B t^2 + C at t = u±a, u±b, u±c), shown as its minimal model. Found by an exhaustive search of every primitive fibre with a<b<c<=200 and u<=200 (about 210 million fibres), ranked by a Nagao-type sum over p<1000. Then the exact minimal-model size was computed and PARI's ellrank was run. The 2-descent upper bound is 13, and the 13 listed points are independent (saturated at primes up to 200, LLL-reduced), so the rank is exactly 13. Minimal discriminant: log|Δ| = 137.627, compared with 140.786 for curve#2158.",
  "created_at": "2026-09-26 20:33:08",
  "updated_at": "2026-09-26 20:33:08"
}