{
  "id": 5290,
  "curve_key": "1005147926983278120023338342437361:-32395282757852570541824417144817130714037420800585",
  "ainvs": [
    "1",
    "0",
    "0",
    "-20940581812151627500486215467445",
    "37494540228996028974655702164236054026654985121"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 228.00419456403844,
  "faltings_height": 16.88691785603784,
  "conductor": "135712212045423145251384547210301605950186052761297293697205002",
  "bad_primes": [
    "2",
    "3",
    "19",
    "23",
    "31",
    "43",
    "79",
    "109",
    "487",
    "95829791",
    "227655920844445350211",
    "424418250966331000111"
  ],
  "discriminant": "-19635951463375909578337760225107225475718071959718082113330054547071643894216794705143756750848",
  "regulator": "36321952605783929195.5788288753456768487971906706749586637555818413705312581175",
  "points": [
    [
      "1655263610727652863/4",
      "-2129483655943492815982738923/8"
    ],
    [
      "15960447651381906",
      "1941384113065954164730071"
    ],
    [
      "-3613088176909170",
      "-256881270745182106642029"
    ],
    [
      "25029173651998644246/9025",
      "-23480860259524900603657009179/857375"
    ],
    [
      "3911985355314072",
      "-124269326501700362747361"
    ],
    [
      "3107858483505506791986/1343281",
      "58880753902397213969773385665617/1556862679"
    ],
    [
      "2651753298501344754102/395641",
      "110799567239185011900554833605711/248858189"
    ],
    [
      "5053067890245522",
      "246379466796375201251799"
    ],
    [
      "2775701525778210",
      "-27480546789250337457369"
    ],
    [
      "10692436586281395/4",
      "-199007837424548000618733/8"
    ],
    [
      "116283014705616317634/38809",
      "310597684395493437373143614523/7645373"
    ],
    [
      "1879658838622171042/729",
      "499078738455176465483525317/19683"
    ],
    [
      "50360707513083954/25",
      "7379904241505740181210907/125"
    ],
    [
      "2705931110919666",
      "25372561970754823098711"
    ],
    [
      "310996448671051282/81",
      "85306355407698246593030591/729"
    ],
    [
      "2685711064987218",
      "-25026783584153277390441"
    ],
    [
      "91123424207306577666/38809",
      "-272548125617074628356793105157/7645373"
    ],
    [
      "4092385239179474",
      "142601935885104975539735"
    ],
    [
      "3937815491011872",
      "126868823369214032477169"
    ],
    [
      "5070508512905010",
      "248350964120058770358231"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber of the x1092 K3 fibration family (file 167_x1092_t0_0085.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -9/19. A(u) = (-1383436870888675706200107)+(-5515600836340711948476888)*u^1+(716527474181252185372992)*u^2+(19599652286308507036273344)*u^3+(-10348738112891504390620680)*u^4+(11954627818857380106162864)*u^5+(-12819504668705779187131920)*u^6+(-9282889399122096275139840)*u^7+(-1414273781478875646923712)*u^8 ; B(u) = (623151887106655251853879329773846694)+(3757259803598750736604123706461347144)*u^1+(3844429892270677596667121714023090128)*u^2+(-16349006985383236996062158267218060800)*u^3+(-30783616351291087563896953575812615016)*u^4+(27168228250111848663166931761100037840)*u^5+(103348463813083983014377415708098752432)*u^6+(-94509325719259964077509140015970425664)*u^7+(-43457863230894151560075098944198688496)*u^8+(25171285331836505034229860335000995200)*u^9+(23456140237510943194372729657306631040)*u^10+(6556407886828091399912763281517445632)*u^11+(637439003051955708260756861597134080)*u^12. Rank exactly 20: 20 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 3 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 20 (kappa=2.8, GRH), rank is exactly 20.",
  "created_at": "2026-10-21 12:16:38",
  "updated_at": "2026-10-21 12:16:38"
}