{
  "id": 5205,
  "curve_key": "137850262130497531915462366356961:-1608709800620975093019953128824937548901717865041",
  "ainvs": [
    "1",
    "0",
    "0",
    "-2871880461052031914905465965770",
    "1861932639607609822042314737359536661699268900"
  ],
  "rank_lower_bound": 21,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 222.01116248655794,
  "faltings_height": 16.34465868000396,
  "conductor": "41501262064916999890778113841582746737819383506183508678951167023510",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "19",
    "23",
    "29",
    "11773499",
    "314541145523949107",
    "323918838634233546559384197955484383"
  ],
  "discriminant": "18275018831081946081795957408618405635206628844358511323546372429488989923309711625088000000",
  "regulator": "47896854683602027640.349537946883721776489282198290541019063987117078231462727422",
  "points": [
    [
      "1074586862602508660/289",
      "-1013930357204742000462655610/4913"
    ],
    [
      "23089117578958521452/22201",
      "-680051014829766128464031786/3307949"
    ],
    [
      "2584002680145964460/529",
      "3930968561898241162268607010/12167"
    ],
    [
      "68245133540065240",
      "17822756085999180489901030"
    ],
    [
      "112045420876537746890/83521",
      "-496764289479238436950761709030/24137569"
    ],
    [
      "1323421584020840",
      "19471099857623463413030"
    ],
    [
      "29728430538367592",
      "-5117603492734996864211854"
    ],
    [
      "11157484538481773435/14884",
      "-20730630820252676939015331185/1815848"
    ],
    [
      "-1875271402100481340/1849",
      "4856689684064001924009056210/79507"
    ],
    [
      "7434223591016960",
      "-625606928730819402239290"
    ],
    [
      "-12839203493630690/9",
      "1492499193427115809522760/27"
    ],
    [
      "109733942375624240/121",
      "2426411409510108674026930/1331"
    ],
    [
      "-5070882669840100705/3844",
      "-13804115603343070922694059585/238328"
    ],
    [
      "13725038147553055508/6889",
      "-36380888947503882788764737758/571787"
    ],
    [
      "85556774348573731460/2209",
      "-790628483335318394954297866310/103823"
    ],
    [
      "1728115725952430",
      "-45385107901095362440360"
    ],
    [
      "944700934777231040/841",
      "178120894709335841667635170/24389"
    ],
    [
      "825329667704300",
      "7339858391534262524870"
    ],
    [
      "823658977395740",
      "7434001794823339736030"
    ],
    [
      "1454893087465940",
      "27626906883870558173030"
    ],
    [
      "1476216185233940",
      "28972529903682521597030"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber 66 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -40/7. A(u) = (-5523027425962212672208603351992987)+(11931822446710335547122692437257792)*u^1+(1691960323687093239746136961856976)*u^2+(-8325704248879965860362076192156544)*u^3+(-5170257801929882674508264302171680)*u^4+(-1860050058415221674720962229101056)*u^5+(-451161875487198592888382302550784)*u^6+(-28773514612260767178393000394752)*u^7+(-699834632413116089430954830592)*u^8 ; B(u) = (159000839890638574975169504773663511028739656150966)+(-501191565495384999221133904426408580841668740373184)*u^1+(197914167405983542084543638913806542299272417123984)*u^2+(428503287351301669607375291308003425413511660240000)*u^3+(-123040423514928090425285111998045420124005272593760)*u^4+(-105923509944757362498894072270789345100066091192832)*u^5+(121741900375944705432887583169914040801120454358528)*u^6+(107629873378225312023391466690280996994598937679872)*u^7+(30533984382625449198465286646323270119548502612480)*u^8+(3821382734656637004860508003894628059868846448640)*u^9+(326817339610894946622160643396035371237311287296)*u^10+(15018242786000481569679654452206674531395567616)*u^11+(280196251925076778185717521863443625575604224)*u^12. Rank 21 (exactly): 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=1e8, exact halving), 21x21 canonical height det = 4.79e20 != 0; Bober analytic rank bound 21 (kappa=2.8, GRH), root number -1, so rank is exactly 21.",
  "created_at": "2026-10-21 09:53:05",
  "updated_at": "2026-10-21 09:53:05"
}