{
  "id": 1931,
  "curve_key": "409089005951407577353544401936:-676030665291079957689387876913796519207419840",
  "ainvs": [
    "0",
    "1",
    "0",
    "-8522687623987657861532175040",
    "782442899642450813837213157949458793913204"
  ],
  "rank_lower_bound": 13,
  "torsion": [
    2
  ],
  "naive_height": 206.44962468724128,
  "faltings_height": 15.25570010831341,
  "conductor": "587067164462035469998813287245201298336901585011216",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "17",
    "19",
    "29",
    "31",
    "43",
    "53",
    "59",
    "71",
    "73",
    "151",
    "421",
    "443",
    "14461",
    "20161",
    "1498361",
    "65188429"
  ],
  "discriminant": "-224858133494881927058583899298762867445188269808784134160154648680743417259552643739648",
  "regulator": "54010090286550.60886559330915630002751629084019540485601072666877",
  "points": [
    [
      "41082966036535566968992535/580066547641",
      "322867751897191131648100802104673230278/441790864080886061"
    ],
    [
      "4017446747162509584411412484684/18873209173280401",
      "-7609542045703596937178619640925495484608557602/2592797682710217660019399"
    ],
    [
      "-597252591996410953771036130486054161181994877/32018236242575946636019292533849",
      "175180495305794350784571875586200919162302600596700297894950617994092/181174097676986150696664687605077932917906393507"
    ],
    [
      "6524040067325853001658987/34151409601",
      "-15620470877415118603010155068786214464/6311214645674401"
    ],
    [
      "6086628120412931732375095034/46386955775209",
      "438142492204399694621338514886675905552808/315932139232926131573"
    ],
    [
      "-3418421087065478572678/31483321",
      "-115535880774357313819692929189976/176652914131"
    ],
    [
      "19120725754533923152418/54479161",
      "-2575545045111528452216584723829280/402110687341"
    ],
    [
      "3022177963265842463922401859424846477043806565/11960605569359339439620022357604",
      "-158925532537223543609743577469886295167151508114586351582809902716209/41364688007866588877468915165855023409947847192"
    ],
    [
      "-3710907119302869034089364/52263903769",
      "12123876901409214242490006917143569174/11948207832342397"
    ],
    [
      "434198699401599290/3721",
      "266333434579837268031304488/226981"
    ],
    [
      "225091243933505977906706492/84716505721",
      "3375084592502807010960182345137017476586/24657670871659981"
    ],
    [
      "1201381509308770910398827636278367128510980200015859060416245179167766933292/4756025957478160252570824561293553786895900221121106998989241",
      "39831475919274377922912363424044366641910427651654628797442817605992783880606750239677290432606560139751195844554/10372091142288945772772381056294491635284704575433767579393120013887015624160764438838240211"
    ],
    [
      "-6855227943667531903/59536",
      "7074650279619162380195956329/14526784"
    ]
  ],
  "submitter": "Alvaro Lozano-Robledo",
  "history": [],
  "commentary": "Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (3,-1,176/9).",
  "created_at": "2026-09-25 19:17:22",
  "updated_at": "2026-09-25 19:17:22"
}