{
  "id": 482,
  "curve_key": "27759608956981987129068202891741675715706927999366620721:-143133322210603745158314389290122480660770237108821460117449302532368174041976557481",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-578325186603791398522254226911284910743894333320137932",
    "165663567373383964303604617378519352971283772210218024456350513024362673656396239"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 382.98953121196143,
  "faltings_height": 29.81494603796802,
  "conductor": "693388196377898328468926745954415305693424993093406004824100867790323941987867548180169417707290",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "23",
    "37",
    "53",
    "61",
    "83",
    "89",
    "127",
    "151",
    "211",
    "337",
    "419",
    "587",
    "673",
    "1031",
    "1231",
    "2269",
    "6229",
    "30631",
    "32579",
    "33797",
    "5297759",
    "5705093",
    "59371564977849024911083"
  ],
  "discriminant": "523317492936369614307068381771829104616573633705456347502700979685996867589584640927466267722534643920024571888036810244540085248838214667263735513764070400000000",
  "regulator": "50765749126560690738.84059239803420460925236068056577305815932278621631",
  "points": [
    [
      "809515092541967941718136597",
      "15099262509653558353555003721799947618101"
    ],
    [
      "-852747653681017412832573003",
      "6223279017945511679406039866082156438901"
    ],
    [
      "30222198675172580753555368484757/78961",
      "13709308142027746022522232834435957063527329901/22188041"
    ],
    [
      "504256587536299622367637413",
      "1502989644611148844965048095854869323653"
    ],
    [
      "535695397238094697279766517",
      "3096077922818629568411741080879273953301"
    ],
    [
      "642119663286090425140003797",
      "7685497482799719534105976848756619730101"
    ],
    [
      "360972897710921637904887477",
      "1984725358592149594600196275971892520821"
    ],
    [
      "668800808126486807996012397",
      "8833467738957341198841580557846078067501"
    ],
    [
      "2257728088463685880622440917",
      "101825151544314607272702568559802386565301"
    ],
    [
      "738335663949270640913225933073/2209",
      "323426453726594807285094744492107422459806623/103823"
    ],
    [
      "-10697228076736515691528799403/16",
      "1018923334834875300943691070060516921679639/64"
    ],
    [
      "13783104989845197131650554679587/19321",
      "28942388196136411144732783844962148325092045269/2685619"
    ],
    [
      "-1551182800578102852063681512147/1849",
      "617882387669644753183223201538726518834186607/79507"
    ],
    [
      "-305760208818297932235124203",
      "17717423280050542804358520891743945378101"
    ],
    [
      "-63851249249720382364549514534627/157609",
      "1142607852056463517156203946713344675523108986073/62570773"
    ],
    [
      "65281936272504730485602740327/81",
      "10888032394802026581466746140177423170880429/729"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=5165/10512 on branch u=19/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).",
  "created_at": "2026-09-01 19:28:15",
  "updated_at": "2026-09-01 19:28:15"
}