{
  "id": 516,
  "curve_key": "78837123528359412966806850981213319106571709905585008881:-696034204494987972299525917756805184128338241452536068417228413932235572119071851721",
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  "rank_lower_bound": 17,
  "torsion": [
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  "naive_height": 386.12093705511137,
  "faltings_height": 30.017731004800495,
  "conductor": "281137155660058521374013654863701954636215183842935698717211507024645830876359657299507174410",
  "bad_primes": [
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    "29",
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    "43",
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  "discriminant": "3201465848473673503881543313334345211840394964094228668724808998319739507096871624539567508802654788245451413523655661131618832483165808164779960129781113536000000",
  "regulator": "17888354866898702478199081.4235530977106194832040878270999497025273378273",
  "points": [
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      "142650668603054232210191723064233435962153225505569440222485870661125/9166984057441718080086530323"
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      "356990400703782082118034496829110528291102501550090/1419154200809975070850369",
      "34163557366122937793825742091669821695863578848647406119279700582013803521705/1690613686930738791973179259177954847"
    ],
    [
      "56026379491391850774286764802740801745803680/45593072103267649",
      "246841148157827553275722057324177120796074931430225529102597107905/9735276359273956173528607"
    ],
    [
      "465371123128969634782814698541088353589655/53176151671204",
      "314237730839837833796162879661473215020663992851756424480042255/387771026864450466392"
    ],
    [
      "-1074776487350171745504871743165339971950/52231942845889",
      "10936650484667254065126377910365582683307186955123316949051545/377488973681695066463"
    ]
  ],
  "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 2). Fiber t=30311/1877 on branch u=31/17 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 13:42:28",
  "updated_at": "2026-09-02 17:33:25"
}