{
  "id": 5161,
  "curve_key": "70738624452164724108458056561:-18757977364310122696955005056754906088924809",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1473721342753431752259542845",
    "21710621949432889570623063437116601159537"
  ],
  "rank_lower_bound": 21,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 199.28636774792056,
  "faltings_height": 14.422228200452523,
  "conductor": "5430668857396047456406533044904719163125021637089857700586223679970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "41",
    "125526913",
    "52557916018672529579",
    "13657686673091233425844408014183493"
  ],
  "discriminant": "1221664636793366424392019216810808381787987683405510515276020788064339430912000000",
  "regulator": "131273110306049777153.78807229212216125921354242052413069507947054898403641757352",
  "points": [
    [
      "-61788887594251/16",
      "-10583102634372458020273/64"
    ],
    [
      "-266109842580567886/6241",
      "-41331330597233679975930463/493039"
    ],
    [
      "-417253195405051/16",
      "13179570443898262255877/64"
    ],
    [
      "-125762521080337/4",
      "-1538120717772794356031/8"
    ],
    [
      "-37168806707686",
      "-158548487234126514457"
    ],
    [
      "-8000821421225884/289",
      "998337913841003551216879/4913"
    ],
    [
      "-18792704479006",
      "206806421805498421583"
    ],
    [
      "-37614636792736",
      "-154675887802562423257"
    ],
    [
      "1207435434546818/49",
      "-6492920936946443944615/343"
    ],
    [
      "8247771328514",
      "100582131434644569743"
    ],
    [
      "-22267393267486",
      "208531930915539297743"
    ],
    [
      "-155225878731649/4",
      "-1144314445637302099211/8"
    ],
    [
      "-528821889825463414/24649",
      "-806695163861584108436026501/3869893"
    ],
    [
      "-41981592896203054/2209",
      "-21492663978428229432305471/103823"
    ],
    [
      "24806682290114",
      "-20440407830173737457"
    ],
    [
      "-1842427235735356/121",
      "268255674019188764513383/1331"
    ],
    [
      "-30933334801036",
      "194161083649795402943"
    ],
    [
      "20611808255042",
      "9561249197299809455"
    ],
    [
      "20471066717714",
      "10985108905239079343"
    ],
    [
      "518572310877266/25",
      "1018368448602662943571/125"
    ],
    [
      "21132340446914",
      "2152411598805244943"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "Fiber 130 of the x1152 K3 fibration (Elkies-Klagsbrun-type family), specialized at u=-5/92. Rank 17 basis from fake 2-descent section lift (mod-p Shioda pairing, det 1152, index 1); extended to 21 independent points via kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e7, exact halving. det of 21x21 canonical height matrix = 1.31e20 != 0. Bober analytic rank bound (kappa=2.8, GRH) = 21, root number -1, so rank is exactly 21.",
  "created_at": "2026-10-21 06:26:22",
  "updated_at": "2026-10-21 06:26:22"
}