{
  "id": 5105,
  "curve_key": "5373492045050113052242204274066641:-392905741214550754640151028052910877608355386103705",
  "ainvs": [
    "1",
    "0",
    "0",
    "-111947750938544021921712589043055",
    "454752015294618919652677037589978355607695572921"
  ],
  "rank_lower_bound": 23,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 233.000358161093,
  "faltings_height": 17.225072625557974,
  "conductor": "77818939565657618194273591520650005763407098459148894438253561617032796993246",
  "bad_primes": [
    "2",
    "3",
    "11",
    "23",
    "41",
    "89",
    "16339",
    "61879",
    "131210951",
    "1826922743",
    "285042858541973941",
    "203362318195937021795561201"
  ],
  "discriminant": "452272729389102478552733380248660085982848248093200135948103384778790446700194498889807473016832",
  "regulator": "170472198673948787018806.308933283717466582969964417518713103465135682059718254252291",
  "points": [
    [
      "47986779759095974/9",
      "-2623335436733874293691889/27"
    ],
    [
      "2184254295688625950/1521",
      "32324739367020860916024959459/59319"
    ],
    [
      "6533826086153424",
      "-47325335068006626604299"
    ],
    [
      "4715692792302506",
      "-178065301078006754138107"
    ],
    [
      "9705826011695358",
      "531530914781777416695093"
    ],
    [
      "6659592785813894",
      "67673428536961517778941"
    ],
    [
      "820353952704562582/729",
      "-11310459782988888247443146281/19683"
    ],
    [
      "5292306030968454",
      "102566209423387157067261"
    ],
    [
      "856156268560134",
      "599612217732033151271421"
    ],
    [
      "1774828890945212982/361",
      "1044853657654737054837694431/6859"
    ],
    [
      "-3971906012175162",
      "914733308813949284563389"
    ],
    [
      "-1760851471081722",
      "803999804885341513449213"
    ],
    [
      "19761560105271966",
      "2441261875218882709566093"
    ],
    [
      "-118918592558451954/25",
      "-117235725569509267366989159/125"
    ],
    [
      "23866385016886311546/3721",
      "-5498803577098714356593950539/226981"
    ],
    [
      "44792707181812031046/9409",
      "-157296435729373121523407624355/912673"
    ],
    [
      "4598102453229318",
      "-192925844228522457159171"
    ],
    [
      "-2877283367586042",
      "867777092558530780413693"
    ],
    [
      "-3841933702153434",
      "910021520692588040892093"
    ],
    [
      "7957491901052646",
      "260404351929736235762493"
    ],
    [
      "4187758191782214",
      "243688229432005284677181"
    ],
    [
      "11908499689247670",
      "900219278867256889033629"
    ],
    [
      "5725480093698918",
      "38536459851399647355453"
    ]
  ],
  "submitter": "hdeping",
  "history": [
    {
      "kind": "primes_recorded",
      "user": "Daniel Błażewicz",
      "at": "2026-10-21 03:25:58"
    }
  ],
  "commentary": "K3 surface X1152 (NS det 1152), 2-neighbor elliptic fibration #118: E(u): y^2 = x^3 + A(u)*x + B(u), deg A = 8, deg B = 12.\nA(u) = (-210004115838928369565529627)*u^0+(229704902521975886120710272)*u^1+(671361220298524562480432016)*u^2+(-825066258136770176365152384)*u^3+(675067726372372360419690720)*u^4+(344099734291872947874737664)*u^5+(-3229034806631787244074342144)*u^6+(978111127560897359180587008)*u^7+(-84883145438575628808198912)*u^8;\nB(u) = (1382777345283637445332465597823493038646)*u^0+(-2141333171588931697572809644883683430784)*u^1+(-10798798391436050750839148859333740178096)*u^2+(14742641160489666238050313976754242805120)*u^3+(41211555355187413879811405147684931157920)*u^4+(-44965174528273847083203586237455710578176)*u^5+(-88399369682746090049036224556561108169216)*u^6+(80287083394355434332739039229066903709696)*u^7+(67485582858387264677580530489530325137920)*u^8+(-61943525190403312492725242979997278904320)*u^9+(29646663812901178376930645566762824781824)*u^10+(-5574761262185546002829095827214624751616)*u^11+(344744578091860653509920340042102562816)*u^12;\nSpecialization u = T = -103/36 gives the listed minimal curve E0, with 17 independent rational sections specialized to the submitted points (Shioda pairing on the fibration: 17x17 integral, diag 4, det 10368, index 3).\nSection formulas (x_i(u), y_i(u)): [[(-10080108258453)*u^0+(17815438478184)*u^1+(33876332297944)*u^2+(-23394199654144)*u^3+(3030849147952)*u^4,(49753553200261911000)*u^0+(-27812028821294922120)*u^1+(-207353738770397240544)*u^2+(-18064297512861905440)*u^3+(162032850478238331264)*u^4+(59094214805600194560)*u^5+(-10738626096765952000)*u^6],[(-10114079969853)*u^0+(-18041207683536)*u^1+(9773675048344104/169)*u^2+(-233552800545216/169)*u^3+(-723222560801360/169)*u^4,(49720819221642254400)*u^0+(-812148566894649776880/13)*u^1+(-3910026452032971237744/13)*u^2+(971878286242884652895808/2197)*u^3+(124771389780554914089024/2197)*u^4+(-425487495033661902605568/2197)*u^5+(55127822923228129851392/2197)*u^6],[(-10211690186658)*u^0+(19053319485528)*u^1+(32208238897380)*u^2+(-25547714357904)*u^3+(5790774069552)*u^4,(-49622710422611232270)*u^0+(25469412856432875846)*u^1+(219012328920022391916)*u^2+(11167380659183666040)*u^3+(-196834385597398478352)*u^4+(-22679768978861618880)*u^5+(6883884071338982400)*u^6],[(-1036582105869)*u^0+(-5439167519328)*u^1+(10762460049960)*u^2+(4471942854528)*u^3+(12798081782832)*u^4,(-39991874723449442640)*u^0+(15687271511145504864)*u^1+(191384899048735843248)*u^2+(-96172978953858924096)*u^3+(-221811111873415038528)*u^4+(119018729616601231872)*u^5+(36805059977813729280)*u^6],[(-10378833990813)*u^0+(-36628014643488)*u^1+(120779033553432)*u^2+(-32082552167904)*u^3+(-1100601991248)*u^4,(-49440521634671550240)*u^0+(87682115251697088672)*u^1+(626760091152730298016)*u^2+(-1408418261085193738272)*u^3+(496483071857128485504)*u^4+(96757910616374455680)*u^5+(-20900572972064755200)*u^6],[(-10504255326525)*u^0+(-376816162752)*u^1+(43233929241336)*u^2+(-17215712091648)*u^3+(6871270155312)*u^4,(-49291796254101500736)*u^0+(46659003193004480640)*u^1+(151010219282731514496)*u^2+(-181965463537307488704)*u^3+(48514484588765898240)*u^4+(-9113408660123698176)*u^5+(-9268879791137218560)*u^6],[(-10614249506093)*u^0+(-29218788720512/3)*u^1+(117764810210552/3)*u^2+(95434591907968/27)*u^3+(-62446118913488/81)*u^4,(-49152698244932922640)*u^0+(59264078619311854912)*u^1+(661333908853560905200/3)*u^2+(-6516624318252378189248/27)*u^3+(-3695939059750605830720/27)*u^4+(13250540893078110479872/81)*u^5+(-14756194666576013593600/729)*u^6],[(-1071363786453)*u^0+(-21381180297336)*u^1+(51924681414360)*u^2+(-9911458355328)*u^3+(3623363679792)*u^4,(40081646882698785000)*u^0+(25311833141056574760)*u^1+(-365068021284392325408)*u^2+(387833030441934942240)*u^3+(-21943442681307812736)*u^4+(-85800067366988911104)*u^5+(9206104486232586240)*u^6],[(-11271974814333)*u^0+(-10090436343552)*u^1+(42113920079736)*u^2+(5601317962752)*u^3+(-1131694783440)*u^4,(48143020827292110720)*u^0+(-6706810337925070",
  "created_at": "2026-10-21 03:24:13",
  "updated_at": "2026-10-21 03:25:58"
}