{
  "id": 5291,
  "curve_key": "6892453529399990079571477294441:-21895567576567229231025533175352525226192001205",
  "ainvs": [
    "1",
    "0",
    "1",
    "-143592781862499793324405776968",
    "25342092102508355199473656337230793829833402"
  ],
  "rank_lower_bound": 21,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 213.40522681461215,
  "faltings_height": 15.783175744424103,
  "conductor": "347225358872056718656732779074095147946415002428488763635566127923802",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "19",
    "23",
    "31",
    "83",
    "103",
    "63044070442806421217",
    "55427010356864589386966709789244988519"
  ],
  "discriminant": "-87953451086090190861214750081495749948550260907471014585442400070033798566310911116298868",
  "regulator": "94517495689343595045.198328595843957787821887510488327201515183774733886362252390",
  "points": [
    [
      "-777510123212470080/2209",
      "589861960861152340390759178/103823"
    ],
    [
      "-96672971205236590722/237169",
      "464143414155728433068612978158/115501303"
    ],
    [
      "-71766784482216119/225",
      "20993567570667944973640159/3375"
    ],
    [
      "-912595527070687386/2209",
      "390607477928453572876734665/103823"
    ],
    [
      "221756919323122647/1156",
      "-86609392685202199731554291/39304"
    ],
    [
      "7461276778937115",
      "-643683228197783380029419"
    ],
    [
      "-90896127449503689/289",
      "-30835392399086723657472631/4913"
    ],
    [
      "-413457410370519",
      "-3745967682384304947995"
    ],
    [
      "-393668820672279",
      "-4567407432509326518395"
    ],
    [
      "1911611948400578493/8464",
      "1639297937809599462234078503/778688"
    ],
    [
      "1096732930583305461/13456",
      "-5877761213316988023793220395/1560896"
    ],
    [
      "23984871932403827/121",
      "2875505999087342634284031/1331"
    ],
    [
      "61529269968172624937244/420373009",
      "23541470643849722580028964353013410/8618907803527"
    ],
    [
      "2697900982215201/25",
      "-416512764346398225370339/125"
    ],
    [
      "-11034014518596021/25",
      "206972788614210096475523/125"
    ],
    [
      "1134303419197006621/144",
      "1206707582381323710534133621/1728"
    ],
    [
      "7191499850319927/49",
      "-934886562034719181743815/343"
    ],
    [
      "357222594910887",
      "4430796649158270633961"
    ],
    [
      "410251022812287",
      "5956565483283074883841"
    ],
    [
      "565506812285049807/1369",
      "306074931922432127758397413/50653"
    ],
    [
      "72607369099181643/289",
      "11121559733425643815250453/4913"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber of the x1092 K3 fibration family (file 222_x1092_t3_0357.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -9/34. A(u) = (-176037064306992808566507)+(5915756000685239016284448)*u^1+(-74739832214501898117378864)*u^2+(407193784801308814869840384)*u^3+(-357852151289462141058677280)*u^4+(-6136035049085750455308401664)*u^5+(20390581978882532274482816256)*u^6+(24085910297771228781276198912)*u^7+(-138434852508586604980011424512)*u^8 ; B(u) = (29353282297132394631805644617637606)+(-1465800220984727623268853753715246176)*u^1+(29684854589829915583090967820635629584)*u^2+(-302504492712046802509595254047692878080)*u^3+(1489300822836997474656472031816649171360)*u^4+(-1859111704531363363401367591512810029568)*u^5+(-8901302556884468646838941329449484892672)*u^6+(7835621959513555079061283351216613480448)*u^7+(5887180595638993725830206009454293117440)*u^8+(905676905416813548887689587645787310653440)*u^9+(-2692953602420758530115438575793251800838144)*u^10+(-4969979756041599501329003211795628338610176)*u^11+(18717651081077897840135894513712398618025984)*u^12. Rank exactly 21: 21 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 4 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e6, exact halving), canonical height Gram det != 0; Bober analytic rank bound 21 (kappa=2.8, GRH), rank is exactly 21.",
  "created_at": "2026-10-21 12:16:43",
  "updated_at": "2026-10-21 12:16:43"
}