{
  "id": 508,
  "curve_key": "2582058539642916008368890300243560980031465480721:-4145162831000512468908935485350397948033349599349835486575635854265278585",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-53792886242560750174351881255074187083988864182",
    "4797642165509852394570540630340076783744661179810401137268571754467525"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 334.41801428427107,
  "faltings_height": 25.63926099664782,
  "conductor": "2824480633135096954485482008313880173502754314875888014401478071245887272764416254",
  "bad_primes": [
    "2",
    "3",
    "7",
    "17",
    "19",
    "31",
    "43",
    "53",
    "61",
    "71",
    "79",
    "131",
    "179",
    "263",
    "1009",
    "2729",
    "2767",
    "3593",
    "29527",
    "135301",
    "762877",
    "1249603",
    "2633311",
    "21043613",
    "80581197209"
  ],
  "discriminant": "18678994192039697456639190949581220336849768420906118062509930672458958527217540760329299428836334039534028340248795738486520823306590093312",
  "regulator": "3645826786026580420239587.07685750327310926337262105310703419073633372035",
  "points": [
    [
      "214654283443990116384063",
      "56047109830495339631998543804198837"
    ],
    [
      "-935816174180979197550056965741166585031/26250819670976209",
      "347357971384075637614457310293749485385697932164930332273741/4253185287699760894270823"
    ],
    [
      "2011139059169064525855030359930823035773617/24103353608979789769",
      "3530601412092410758989455975820106661320404441100107312449308247/118335815550780270467667886853"
    ],
    [
      "741037380176440388212870502711654761/5701020557761",
      "17318895712939965606469553358533851457867541381661067/13612218466375342241"
    ],
    [
      "8658888625985292707426253923450073/40130906929",
      "457319295407546475809820757156982748061271120664221/8039304192365783"
    ],
    [
      "17173934981470959373051777879017/132733441",
      "2905710959616661362673536128052902177924197675/1529221973761"
    ],
    [
      "-11954133171633191227816246777676655/2653448781481",
      "306850471568829589573794813695615866887155339437100935/4322311511554441621"
    ],
    [
      "6985868630425762971677974847016782121/48209345457025",
      "2256365193808282702621187063792754353876148321239765909/334731707265034397375"
    ],
    [
      "37201949704061014023415812481917017433/261149252172769",
      "21300474946917688757104575063486566251251223591635976555/4220201424977442562897"
    ],
    [
      "8980038470842565894980915906227415557201/69479571593268841",
      "37155314071950776952556763579323143423766160590469754678809/18314104921284411686776139"
    ],
    [
      "5549376664773102652715645176017169456239201/42525900613303810921",
      "101609782675025274479902470453598769642102051468861292778338743/277319417816424913368238780469"
    ],
    [
      "33145021988186079265571608508910962425/258614031153361",
      "12163419260233643993793502984257534213154283056806974259/4158896628326183007641"
    ],
    [
      "1076713607012983322491488093744201/7456495201",
      "4116815484452726414003389818220235032981403021861/643875817101551"
    ],
    [
      "-1899964530443937372841299608453183241/10424277682225",
      "3111603990428640999184461865347348257513415795177005093/33656500502880979625"
    ],
    [
      "-493444943963856306114291618755542835/2799594893601",
      "439508421030790023937434770669286883690614378412808305/4684279376378299599"
    ],
    [
      "844180456472167124814463676999254662297/3372257649326449",
      "16406317109771456253864592796905717490934384735432365223293/195830856186880073561207"
    ],
    [
      "31120677685172612816823710643230600208288161514601/238827059197155333453030001",
      "2989659702837021538250608604148943262345565217828335698184600084083328739/3690840660938844921315372966147314986249"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 1). Fiber t=431/1490 on branch u=1/17 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).",
  "created_at": "2026-09-02 13:41:34",
  "updated_at": "2026-09-02 17:33:23"
}