{
  "id": 505,
  "curve_key": "119894277589030646200728698321381937529308842425:-40045683856562421478067618027101666001543855402621990484750558206558125",
  "ainvs": [
    "1",
    "0",
    "1",
    "-2497797449771471795848514548362123698527267551",
    "46349171130280580414429905307506113953211289710155681032227157688998"
  ],
  "rank_lower_bound": 18,
  "torsion": [
    2
  ],
  "naive_height": 325.2088185572892,
  "faltings_height": 25.02352610165016,
  "conductor": "114713100997610410312595038097570863411319285752217220984538901104669511816895282450",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "23",
    "83",
    "113",
    "131",
    "139",
    "157",
    "181",
    "383",
    "461",
    "569",
    "1721",
    "2381",
    "2521",
    "4049",
    "8117",
    "13697",
    "226769",
    "340757",
    "522157",
    "844717",
    "1973893",
    "2830897"
  ],
  "discriminant": "69317140582662900877610095614907960786971423401250978324706785878258678236968916883842023737085086042782754420373162513987494823429937500",
  "regulator": "55619160923131104523648063.499729584626965197303552507589585354316886444213",
  "points": [
    [
      "8046669332028274407152",
      "5174092565810755779402316655974386"
    ],
    [
      "29257586126160441216460713169556623607/115159275850441",
      "155390824282337764874406193474544069819287286321454494454/1235800560625252121989"
    ],
    [
      "1441253172829817864212936489134578/63704245609",
      "19152400571715159357184775609442080363128196656368/16078760478974773"
    ],
    [
      "101220127608732761787752",
      "28819681688301943479648893491860186"
    ],
    [
      "836591976494771131026464508038372/157616146081",
      "360776071120953020414939547829914112499509728825774/62575028539471729"
    ],
    [
      "1822493742923123957768393356583/84309124",
      "1213184373681653061457104454939517513089161963/774126376568"
    ],
    [
      "24314347157433471439372633160245529713463402/1739970703576819081",
      "119892259737335277923238501083081162081409462462074784679090172136/2295159670029022392764567771"
    ],
    [
      "794407329000130996778859332868744316493/32913863346899089",
      "2091762563822303042719868498660068517460134716962646068972/5971291773812780247455687"
    ],
    [
      "62657284861968231220003068282128947025999258/6784820729711320423009",
      "2741850574090631833310915239735762720701378879662014653893636452488/558865841233048966842317203690223"
    ],
    [
      "1891309600024177268569432701077/54213769",
      "515631816725770949394711599240774547884258538/399175981147"
    ],
    [
      "2118433284028401613445352877372664912/39758687091601",
      "2013846656180349863279597856587343270869021004174423669/250696373763048433849"
    ],
    [
      "21619782409441843477285825679597/203889841",
      "90847939897635729536697785816632438462072065324/2911343039639"
    ],
    [
      "18521404772795566261228124593/762129",
      "1977182377523897666991056927568125412722/665338617"
    ],
    [
      "1280419055380765168940936457593/67683529",
      "1348729589974914016870000821797547058236867828/556832393083"
    ],
    [
      "-11545597503491542763505183374325935050242967/1556965291338350647009",
      "493260107455945210891428021649834210691162014433441666836663047012/61435358477933425374779654036527"
    ],
    [
      "17776404705665020708952837063963/309126724",
      "52378372398205007287759007712653285800037558733/5435066061368"
    ],
    [
      "69114581985415856170174396536557271350071/7041437300684248900",
      "89176185172926351249824561033071681148959936526635603966061659/18684951495739693812019763000"
    ],
    [
      "146026031744432562803108110311840498164399177/2214722655764177523721",
      "1352122879165930910261535498157379352674194672740981740055833245014/104226708412194954173815356693019"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 18, unconditional: rational 2-torsion point, 18 independent points (Neron-Tate height matrix of rank 18; 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 18 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=-10949/24231 on branch u=29/13 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 13:41:21",
  "updated_at": "2026-09-02 13:41:21"
}