{
  "id": 2165,
  "curve_key": "51038306501872648865019856:-361402275275830076702063850937020134080",
  "ainvs": [
    "0",
    "1",
    "0",
    "-1063298052122346851354580",
    "418289670457782230165570156300729184"
  ],
  "rank_lower_bound": 13,
  "torsion": [
    2
  ],
  "naive_height": 177.58385607156276,
  "faltings_height": 12.658469474061402,
  "conductor": "294986299695893757693379720933488412787544",
  "bad_primes": [
    "2",
    "3",
    "7",
    "37",
    "41",
    "43",
    "61",
    "127",
    "163",
    "179",
    "199",
    "239",
    "277",
    "283",
    "607",
    "1097",
    "17047"
  ],
  "discriminant": "1353313441060786486966131853463742562426551872215180871005207459445129472",
  "regulator": "360238448883.1915061719904595963833477668928484999174471829121187",
  "points": [
    [
      "901172968614",
      "438096152073226902"
    ],
    [
      "778671108060",
      "249921656585530272"
    ],
    [
      "222507029454",
      "438992669778857262"
    ],
    [
      "315585665016",
      "337873622441105016"
    ],
    [
      "1375220545071",
      "1247750304897931866"
    ],
    [
      "1962945007614",
      "2427884205308227902"
    ],
    [
      "211548730056",
      "450352894580771616"
    ],
    [
      "663336961710",
      "69595961180265522"
    ],
    [
      "917647487535",
      "463989920651450022"
    ],
    [
      "762524981574",
      "225530891982612342"
    ],
    [
      "-12145301853521700/18769",
      "2350201771063996954741584/2571353"
    ],
    [
      "4171211844707256/3481",
      "190984840514455092185736/205379"
    ],
    [
      "20416896231772974/24025",
      "1334432866710684334315482/3723875"
    ]
  ],
  "submitter": "Steps Unbounded",
  "history": [],
  "commentary": "Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=-9/13; one of the smallest fibres of the family (|a|,b<=40 over 8 values of u), rank certified with PARI ellrank: 13 independent points.",
  "created_at": "2026-09-26 14:17:36",
  "updated_at": "2026-09-26 14:17:36"
}