{
  "id": 527,
  "curve_key": "362047504264632941481134761345706658126559286332091379825:-6779673481855600840968327061771901768497961987472852893744863737907471683669884975625",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-7542656338846519614190307528035555377636651798585237080",
    "7846844307703241714083711878936476316769456277937452916435814466987255176116083547"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 390.6941113564352,
  "faltings_height": 30.443404383232224,
  "conductor": "43474516028660989841423831580088508738112258305978774012023297879965482051705634517491450",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "19",
    "23",
    "29",
    "37",
    "83",
    "127",
    "139",
    "199",
    "239",
    "461",
    "499",
    "523",
    "2687",
    "11287",
    "101063",
    "139753",
    "166667",
    "739111",
    "866311",
    "3589133",
    "15017903",
    "1560594379"
  ],
  "discriminant": "863792463438850810499813470881430826697895035564996350280399616447126620659706126257255255657128934440745210076146169227071393499352344116440534001610452317696000000",
  "regulator": "3541884225448180621905489.696566577508788510470906772182694585642486608",
  "points": [
    [
      "92940306327734770762128439177457302244285/18110190938449",
      "24889731165997848983516480048536212599944035891578414535440223/77069855329000133543"
    ],
    [
      "2090654673328086302410257223811570461/1075380849",
      "812598580625623227177906068206999726428444502192757585/35264964181257"
    ],
    [
      "101678767598580702295165163543215474829/46877013121",
      "417406457100652148345488700658063296871909730411821897975/10149388987840831"
    ],
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      "28521341687361713996721521075474502701/16199434729",
      "10239573424128010612142779015372294410652232812776741645/2061815454002933"
    ],
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      "7077414621310719774794854690780053501/3877926529",
      "3054755764441133895789608200540327298234148585947834905/241490118740417"
    ],
    [
      "1346913455938096450860181479850326782905377513589709/1042779407036766661034161",
      "17146772383727510280972531516608146699730325569629073576811397070569780651665/1064850573435988074081903119156244841"
    ],
    [
      "175692747284836873455820352773636415836117/86636760404769",
      "24066807462404931002037789653358699881855523498522220762924255/806405175893664113103"
    ],
    [
      "8128852910775210377772898773951013315189759831709/43811547431310114481",
      "23173764135804277927279282750426233960808382244889531747389756461328911055/289989911350722715123290814871"
    ],
    [
      "-1074294045215520663283959833759301889499/339489179649",
      "1034971291783433955627076362635198426272919943287383227425/197805746946747393"
    ],
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      "1963527626836207925931829821724508498837/1036684403329",
      "19899584269363448412988933987658725954332986949119795799825/1055528215728311233"
    ],
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      "1277036701014875623300545788287972711214829061/432345284026080489",
      "30270116920981433884467051548163086709100701598802788898541741902275/284279857552213667664654987"
    ],
    [
      "39646052283259200587873062983655361393228878181/28404361291874569369",
      "935980733110694194553886015494885895944261201226432355309793792055525/151383151477910071339502081597"
    ],
    [
      "1726362226669440056354070553300877595071069765/919860723461502601",
      "15318361241028769519704789885073492601194558672666544356510995766227/882232623829508613691211099"
    ],
    [
      "29556395063897067832755102804893896278908021/10312117146474025",
      "103527418792160928863026755535499098567175547399004707178148134659/1047181007699846442744875"
    ],
    [
      "843693060945041467977571769727572321/112296409",
      "728984728480064987949607950336293071695702612503861745/1190005046173"
    ],
    [
      "-183596488959116181752821131624452561875169585091/238283673235930846225",
      "422615130314897184402013524105136171093683929230553907176590348429346753/3678251573655062606176307061625"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 16, unconditional: rational 2-torsion point, 16 independent points (Neron-Tate height matrix of rank 16; 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 16 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=-24179/8214 on branch u=38/17 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 13:43:32",
  "updated_at": "2026-09-02 17:33:28"
}