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Properties

Label 32.0.56950036795...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 17^{24}$
Root discriminant $79.18$
Ramified primes $2, 5, 17$
Class number $1920$ (GRH)
Class group $[2, 8, 120]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191844571, -165908186, 801634283, -566599754, 1558431493, -671119730, 1859984222, -146741502, 1519072624, 431086826, 875287303, 513143686, 336933634, 270739944, 67877410, 71151504, -2344899, 3621240, -3693434, -3581160, 77200, -1233434, 395263, -207118, 106186, -22782, 13646, -1874, 1051, -98, 47, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 47*x^30 - 98*x^29 + 1051*x^28 - 1874*x^27 + 13646*x^26 - 22782*x^25 + 106186*x^24 - 207118*x^23 + 395263*x^22 - 1233434*x^21 + 77200*x^20 - 3581160*x^19 - 3693434*x^18 + 3621240*x^17 - 2344899*x^16 + 71151504*x^15 + 67877410*x^14 + 270739944*x^13 + 336933634*x^12 + 513143686*x^11 + 875287303*x^10 + 431086826*x^9 + 1519072624*x^8 - 146741502*x^7 + 1859984222*x^6 - 671119730*x^5 + 1558431493*x^4 - 566599754*x^3 + 801634283*x^2 - 165908186*x + 191844571)
 
gp: K = bnfinit(x^32 - 2*x^31 + 47*x^30 - 98*x^29 + 1051*x^28 - 1874*x^27 + 13646*x^26 - 22782*x^25 + 106186*x^24 - 207118*x^23 + 395263*x^22 - 1233434*x^21 + 77200*x^20 - 3581160*x^19 - 3693434*x^18 + 3621240*x^17 - 2344899*x^16 + 71151504*x^15 + 67877410*x^14 + 270739944*x^13 + 336933634*x^12 + 513143686*x^11 + 875287303*x^10 + 431086826*x^9 + 1519072624*x^8 - 146741502*x^7 + 1859984222*x^6 - 671119730*x^5 + 1558431493*x^4 - 566599754*x^3 + 801634283*x^2 - 165908186*x + 191844571, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} + 47 x^{30} - 98 x^{29} + 1051 x^{28} - 1874 x^{27} + 13646 x^{26} - 22782 x^{25} + 106186 x^{24} - 207118 x^{23} + 395263 x^{22} - 1233434 x^{21} + 77200 x^{20} - 3581160 x^{19} - 3693434 x^{18} + 3621240 x^{17} - 2344899 x^{16} + 71151504 x^{15} + 67877410 x^{14} + 270739944 x^{13} + 336933634 x^{12} + 513143686 x^{11} + 875287303 x^{10} + 431086826 x^{9} + 1519072624 x^{8} - 146741502 x^{7} + 1859984222 x^{6} - 671119730 x^{5} + 1558431493 x^{4} - 566599754 x^{3} + 801634283 x^{2} - 165908186 x + 191844571 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5695003679558247880375471569828315136000000000000000000000000=2^{48}\cdot 5^{24}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.18$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(259,·)$, $\chi_{680}(137,·)$, $\chi_{680}(523,·)$, $\chi_{680}(273,·)$, $\chi_{680}(387,·)$, $\chi_{680}(89,·)$, $\chi_{680}(409,·)$, $\chi_{680}(667,·)$, $\chi_{680}(33,·)$, $\chi_{680}(169,·)$, $\chi_{680}(171,·)$, $\chi_{680}(307,·)$, $\chi_{680}(441,·)$, $\chi_{680}(443,·)$, $\chi_{680}(123,·)$, $\chi_{680}(577,·)$, $\chi_{680}(67,·)$, $\chi_{680}(203,·)$, $\chi_{680}(531,·)$, $\chi_{680}(81,·)$, $\chi_{680}(339,·)$, $\chi_{680}(217,·)$, $\chi_{680}(353,·)$, $\chi_{680}(611,·)$, $\chi_{680}(361,·)$, $\chi_{680}(497,·)$, $\chi_{680}(579,·)$, $\chi_{680}(659,·)$, $\chi_{680}(489,·)$, $\chi_{680}(633,·)$, $\chi_{680}(251,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{177} a^{20} + \frac{20}{177} a^{18} + \frac{1}{59} a^{16} - \frac{2}{59} a^{15} + \frac{25}{59} a^{14} + \frac{20}{59} a^{13} - \frac{1}{59} a^{12} - \frac{15}{59} a^{11} + \frac{47}{177} a^{10} - \frac{22}{59} a^{9} + \frac{27}{59} a^{8} + \frac{24}{59} a^{7} + \frac{19}{59} a^{6} - \frac{3}{59} a^{5} + \frac{8}{59} a^{4} - \frac{13}{59} a^{3} - \frac{25}{177} a^{2} - \frac{1}{59} a + \frac{79}{177}$, $\frac{1}{177} a^{21} + \frac{20}{177} a^{19} + \frac{1}{59} a^{17} - \frac{2}{59} a^{16} + \frac{25}{59} a^{15} + \frac{20}{59} a^{14} - \frac{1}{59} a^{13} - \frac{15}{59} a^{12} + \frac{47}{177} a^{11} - \frac{22}{59} a^{10} + \frac{27}{59} a^{9} + \frac{24}{59} a^{8} + \frac{19}{59} a^{7} - \frac{3}{59} a^{6} + \frac{8}{59} a^{5} - \frac{13}{59} a^{4} - \frac{25}{177} a^{3} - \frac{1}{59} a^{2} + \frac{79}{177} a$, $\frac{1}{177} a^{22} - \frac{43}{177} a^{18} - \frac{2}{59} a^{17} + \frac{5}{59} a^{16} + \frac{1}{59} a^{15} - \frac{29}{59} a^{14} - \frac{2}{59} a^{13} - \frac{70}{177} a^{12} - \frac{17}{59} a^{11} + \frac{26}{177} a^{10} - \frac{8}{59} a^{9} + \frac{10}{59} a^{8} - \frac{11}{59} a^{7} - \frac{18}{59} a^{6} - \frac{12}{59} a^{5} + \frac{26}{177} a^{4} + \frac{23}{59} a^{3} + \frac{16}{59} a^{2} + \frac{20}{59} a + \frac{13}{177}$, $\frac{1}{177} a^{23} - \frac{43}{177} a^{19} - \frac{2}{59} a^{18} + \frac{5}{59} a^{17} + \frac{1}{59} a^{16} - \frac{29}{59} a^{15} - \frac{2}{59} a^{14} - \frac{70}{177} a^{13} - \frac{17}{59} a^{12} + \frac{26}{177} a^{11} - \frac{8}{59} a^{10} + \frac{10}{59} a^{9} - \frac{11}{59} a^{8} - \frac{18}{59} a^{7} - \frac{12}{59} a^{6} + \frac{26}{177} a^{5} + \frac{23}{59} a^{4} + \frac{16}{59} a^{3} + \frac{20}{59} a^{2} + \frac{13}{177} a$, $\frac{1}{708} a^{24} - \frac{1}{354} a^{21} + \frac{77}{177} a^{19} + \frac{167}{708} a^{18} - \frac{15}{59} a^{17} + \frac{9}{118} a^{16} - \frac{5}{59} a^{15} - \frac{82}{177} a^{14} - \frac{10}{59} a^{13} + \frac{341}{708} a^{12} + \frac{62}{177} a^{11} + \frac{1}{3} a^{10} - \frac{2}{59} a^{9} + \frac{23}{59} a^{8} - \frac{20}{59} a^{7} - \frac{337}{708} a^{6} - \frac{1}{59} a^{5} - \frac{43}{118} a^{4} + \frac{13}{354} a^{3} - \frac{29}{59} a^{2} + \frac{61}{177} a - \frac{143}{708}$, $\frac{1}{708} a^{25} - \frac{1}{354} a^{22} + \frac{167}{708} a^{19} + \frac{8}{177} a^{18} + \frac{9}{118} a^{17} - \frac{23}{59} a^{16} + \frac{26}{177} a^{15} + \frac{12}{59} a^{14} + \frac{269}{708} a^{13} - \frac{61}{177} a^{12} - \frac{16}{177} a^{11} - \frac{85}{177} a^{10} + \frac{6}{59} a^{9} + \frac{25}{59} a^{8} + \frac{143}{708} a^{7} + \frac{11}{59} a^{6} - \frac{53}{118} a^{5} - \frac{143}{354} a^{4} + \frac{28}{59} a^{3} + \frac{13}{59} a^{2} + \frac{73}{708} a - \frac{65}{177}$, $\frac{1}{71508} a^{26} + \frac{43}{71508} a^{25} + \frac{13}{35754} a^{24} - \frac{71}{35754} a^{23} + \frac{13}{35754} a^{22} - \frac{10}{17877} a^{21} - \frac{77}{71508} a^{20} + \frac{15109}{71508} a^{19} - \frac{2263}{5959} a^{18} + \frac{1583}{11918} a^{17} - \frac{3361}{17877} a^{16} + \frac{58}{303} a^{15} - \frac{10225}{23836} a^{14} - \frac{201}{404} a^{13} + \frac{8459}{35754} a^{12} + \frac{8467}{17877} a^{11} - \frac{1849}{5959} a^{10} + \frac{1788}{5959} a^{9} + \frac{35495}{71508} a^{8} - \frac{19351}{71508} a^{7} - \frac{3368}{17877} a^{6} - \frac{6104}{17877} a^{5} - \frac{263}{606} a^{4} - \frac{8315}{17877} a^{3} + \frac{14225}{71508} a^{2} - \frac{21013}{71508} a - \frac{1441}{11918}$, $\frac{1}{71508} a^{27} - \frac{5}{71508} a^{25} - \frac{4}{5959} a^{24} + \frac{6}{5959} a^{23} + \frac{9}{11918} a^{22} + \frac{9}{23836} a^{21} - \frac{41}{17877} a^{20} - \frac{9233}{71508} a^{19} - \frac{17801}{35754} a^{18} - \frac{7919}{35754} a^{17} + \frac{532}{5959} a^{16} - \frac{3563}{71508} a^{15} - \frac{914}{5959} a^{14} + \frac{23851}{71508} a^{13} - \frac{3289}{11918} a^{12} + \frac{8314}{17877} a^{11} - \frac{2959}{17877} a^{10} - \frac{5989}{71508} a^{9} + \frac{1689}{5959} a^{8} + \frac{20519}{71508} a^{7} - \frac{725}{5959} a^{6} - \frac{881}{11918} a^{5} - \frac{2505}{11918} a^{4} - \frac{8449}{23836} a^{3} - \frac{2632}{17877} a^{2} + \frac{10355}{71508} a - \frac{17323}{35754}$, $\frac{1}{71508} a^{28} - \frac{35}{71508} a^{25} + \frac{38}{17877} a^{23} + \frac{157}{71508} a^{22} + \frac{10}{17877} a^{21} + \frac{13}{11918} a^{20} - \frac{19243}{71508} a^{19} + \frac{1219}{5959} a^{18} + \frac{3727}{11918} a^{17} - \frac{487}{71508} a^{16} - \frac{2398}{17877} a^{15} + \frac{5393}{17877} a^{14} + \frac{4625}{23836} a^{13} - \frac{3818}{17877} a^{12} + \frac{2915}{17877} a^{11} - \frac{9215}{23836} a^{10} + \frac{1034}{5959} a^{9} - \frac{4977}{11918} a^{8} + \frac{9887}{71508} a^{7} + \frac{1134}{5959} a^{6} + \frac{10427}{35754} a^{5} - \frac{7201}{71508} a^{4} + \frac{6596}{17877} a^{3} - \frac{1795}{5959} a^{2} - \frac{32045}{71508} a + \frac{3080}{17877}$, $\frac{1}{71508} a^{29} - \frac{5}{35754} a^{25} - \frac{49}{71508} a^{24} + \frac{35}{71508} a^{23} - \frac{5}{5959} a^{22} + \frac{23}{17877} a^{21} - \frac{61}{35754} a^{20} + \frac{4829}{35754} a^{19} - \frac{3033}{23836} a^{18} - \frac{27415}{71508} a^{17} - \frac{4373}{11918} a^{16} - \frac{93}{5959} a^{15} - \frac{2251}{35754} a^{14} + \frac{16823}{35754} a^{13} + \frac{28999}{71508} a^{12} + \frac{5527}{71508} a^{11} - \frac{960}{5959} a^{10} + \frac{3225}{11918} a^{9} - \frac{992}{5959} a^{8} - \frac{2023}{35754} a^{7} + \frac{17065}{71508} a^{6} - \frac{24875}{71508} a^{5} + \frac{2607}{11918} a^{4} - \frac{12979}{35754} a^{3} + \frac{13543}{35754} a^{2} + \frac{8345}{35754} a - \frac{3203}{23836}$, $\frac{1}{71508} a^{30} - \frac{23}{71508} a^{25} - \frac{2}{17877} a^{24} + \frac{34}{17877} a^{23} - \frac{13}{17877} a^{22} + \frac{7}{5959} a^{21} - \frac{16781}{71508} a^{19} + \frac{226}{5959} a^{18} - \frac{1673}{11918} a^{17} - \frac{4451}{35754} a^{16} + \frac{8821}{35754} a^{15} - \frac{40}{177} a^{14} + \frac{34405}{71508} a^{13} - \frac{3869}{17877} a^{12} - \frac{1232}{17877} a^{11} + \frac{9431}{35754} a^{10} - \frac{1090}{5959} a^{9} - \frac{1664}{5959} a^{8} + \frac{9395}{71508} a^{7} + \frac{8354}{17877} a^{6} - \frac{1139}{35754} a^{5} + \frac{2533}{17877} a^{4} + \frac{2871}{5959} a^{3} + \frac{8324}{17877} a^{2} + \frac{9329}{71508} a - \frac{12427}{71508}$, $\frac{1}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{31} + \frac{452531046592249463973809386829933952972551121544939537355663679005072498205229573904405024935590064725699218179}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{30} + \frac{3094202911821416434526347800700353910633731913455161111260381187253905178395008774770933201520186014625562942}{21894063096591091563054925873241618382904495879911911189994356129513012687413520166441095116146734475101106107613273} a^{29} + \frac{1227559561322411305837099373988028670262528997389902080600057143109733005487707812360520256206692289898612483}{1484343260785836716139317008355363958163016669824536351864024144373763572028035265521430177365880303396685159838188} a^{28} + \frac{173761106694859823005471705261892592675039066984935570061122459927139216618181167226911605324084733690024878057}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{27} + \frac{28053031927692562245726733663918806961021366663359287742925736164495270872900537059520831410808216767564724313}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{26} - \frac{16178181337227647842294061161662046864948776993287860333574041982090543823799091150779075007559530341929832052387}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{25} + \frac{1278354926980430044461625126735517963800380717999102309908562875830976775094580782547982463611042237698419215826}{7298021032197030521018308624413872794301498626637303729998118709837670895804506722147031705382244825033702035871091} a^{24} + \frac{33712560773732113613785177732088870421381231654364845859232492312739400000361504119352728940333330563472689159849}{21894063096591091563054925873241618382904495879911911189994356129513012687413520166441095116146734475101106107613273} a^{23} + \frac{113531258458241469775460266035043841587583725009583742897693233708532980697096608472577497211520891458079517045523}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{22} + \frac{249419922235133904313916717842825651729945423190205033777138699949962163106757895798256596763714841502191132631}{494781086928612238713105669451787986054338889941512117288008048124587857342678421840476725788626767798895053279396} a^{21} + \frac{99387651934357777267592789767371739642225497396151349941373272630963367454956066473532764235036732822642771475569}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{20} + \frac{32863517201763265232529665404803390164474021696344894860945678198576673467262515109940540996375744606068347671307511}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{19} - \frac{2275052427098600017109685213203531022644229352030634105851610290113150785615774732198491593925390934567745150727054}{7298021032197030521018308624413872794301498626637303729998118709837670895804506722147031705382244825033702035871091} a^{18} + \frac{4730068135798227563578660713299456400437281681226160058788641476525622305712005387639496820373574323511927535806074}{21894063096591091563054925873241618382904495879911911189994356129513012687413520166441095116146734475101106107613273} a^{17} + \frac{2207508276193162149178690546202744516922818816961749240144589527745021243614160642930551805015737541245147782141259}{29192084128788122084073234497655491177205994506549214919992474839350683583218026888588126821528979300134808143484364} a^{16} + \frac{10576575942844892843655254883121207182785981218936548661624708107823128251390966561398272490841038343431699745763245}{29192084128788122084073234497655491177205994506549214919992474839350683583218026888588126821528979300134808143484364} a^{15} - \frac{7765378660812577761937513672633193643703063290031929374441474693945258745558340266421127779909390268562444125720961}{29192084128788122084073234497655491177205994506549214919992474839350683583218026888588126821528979300134808143484364} a^{14} - \frac{33750101010996925188969451999052744104333324160592115619158935829111744504302464933522789107107767367976261851332525}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{13} + \frac{2248995555644237528591117189040351329291493647408418897223975543949260402754800519154079223607652947362709382463211}{43788126193182183126109851746483236765808991759823822379988712259026025374827040332882190232293468950202212215226546} a^{12} + \frac{12462757091247699488890081657704469002753971202007790961234019810033350563699910286856721952364074732819619223449455}{43788126193182183126109851746483236765808991759823822379988712259026025374827040332882190232293468950202212215226546} a^{11} + \frac{9408124884171069579033122759444752312081872944165172414407868061552239480645978605392898822650516528281261400120129}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{10} + \frac{9251143409070858046914147000662448015142478194742970358691307683638680896718868127898177423892084648970271179975815}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{9} - \frac{28421184571724626444864457908071727546402642725692214761221322918406783749264857399519495998132543452014466053955623}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{8} - \frac{43169146383537356225850252235339206158282869624945854493450188755239045046472479157282066482327977854332079097350799}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{7} + \frac{2638579293101922646855607832458568763320744315834947586949920955647232166710384434432502821012854471149340165660731}{14596042064394061042036617248827745588602997253274607459996237419675341791609013444294063410764489650067404071742182} a^{6} - \frac{8709747346471045525965874039357355239483334434518178873575094202284158024967232916355700972859009090684715304772671}{43788126193182183126109851746483236765808991759823822379988712259026025374827040332882190232293468950202212215226546} a^{5} + \frac{9827121189259053605218311239474938894921767367844803394839299496693794536128581863929983160549782099292725545590445}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{4} - \frac{30531524308834403712987421147551421213202971968123224394487365999540785253971727635787546741533979056402449410901447}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092} a^{3} - \frac{9809715626425888845776647309412359164006004689829052346713074353384461259243932762093674126583753715613518815663237}{29192084128788122084073234497655491177205994506549214919992474839350683583218026888588126821528979300134808143484364} a^{2} - \frac{10753383308219101840245976330650621422821745640051146069243545320932712631041263835219305502071758013381129211407662}{21894063096591091563054925873241618382904495879911911189994356129513012687413520166441095116146734475101106107613273} a + \frac{5847538285837320535245933904259516962862432850026885471760600520773728047688991098225743071612022629858416956858249}{87576252386364366252219703492966473531617983519647644759977424518052050749654080665764380464586937900404424430453092}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{8}\times C_{120}$, which has order $1920$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{203582885106549683845427159919965612000220955315751309221171604936318987712577532359792}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{31} - \frac{85196170518830830766678610644206060947585793644923132329239976635191029371538452299840}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{30} - \frac{26826132538205179217106877216199722405317165813635813892426581450877855992288933458142744}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{29} + \frac{1037451817080930739777210482343747019005631275196141262813267250251382877532243106711804}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{28} - \frac{547736267567534740928434359733389716121691502310325925895223474602401939860719934730633384}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{27} - \frac{36976837823034823132248258116429228389968100550921393266882510318245813440731888581091056}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{26} - \frac{112472893829869231290181825412362526529885004456589113771496799988072875569573631407867920}{394522470187330651737404949200510035087315929383370139280415004642718411457409571774173697937} a^{25} - \frac{1688406723459652556453875268861093353813096371939682350854628370840445628678925059803712655}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{24} - \frac{44333814094045961527207467919779751370972965497287300154243650794415520120148671966114699112}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{23} + \frac{13687876453492581111719067673746315882553047176713799409974839078505253809420901514485428828}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{22} - \frac{38639253875191927906590377577627548667724697616136804797667553092964084024731226919535523728}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{21} + \frac{145627459115403867737008797936296854299819970109202118039583610522566500766675010865590230864}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{20} + \frac{1301883098806810950813855998522875996255504874101597061131862129917732820047106039018454066088}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{19} + \frac{948105722782763864504029758794929119852671664678936213758814392957001697869669955898355142292}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{18} + \frac{6351913584608522170022790068217787520510871702882969629199919081003601882129349368776433279656}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{17} + \frac{2496487318953571913048716913579140261607215256894701302608757687323599262032016287129514171860}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{16} - \frac{1533054088696258694667351713870702129100798206451923493557716318651557600301411385506401576936}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{15} - \frac{51449848112403728331242115661322873677482438832037891339900608117581307348341969000430045304368}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{14} - \frac{134540467663294578265202195474919993463034187191301087523724940977991849902100901615108694285528}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{13} - \frac{283264431425018260027955867952060593502526949062847544780162664447845439909192283741442820551388}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{12} - \frac{527267456218132060441853790126770006832819853327918335018018581703619329718856707243143993971048}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{11} - \frac{719156278966907008308602481090447276637928517313006176083150279032048677785759376914581414340492}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{10} - \frac{1048863381855889269865183174216330175603018485474595744554464182046212547889878429358626771553600}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{9} - \frac{352968892728164043046883169798681680788892424271787987990972737482421131109249517469019488707608}{7758941913684169484168964000943364023383879944539612739181495091306795425329054911558749392761} a^{8} - \frac{1182584452745060468925180487417664537557409720587693091562450954923046862221326531160055072976848}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{7} - \frac{1039193011712669210943829953479100030364261698338968825966319009244301003415271736999557843944900}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{6} - \frac{710121613740373794213874872122628787046471253778012691300370924293347572152852059369347219036584}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{5} - \frac{872863213190820389005201149808420285369259798824808409907632933044469049103265725386968565101356}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{4} - \frac{144788051513007176153550460126246031881132815107108757446968056657894501775639389655456418919728}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{3} - \frac{684222669421070610254034532532754868873862684725800875478753561795349686343165667071011969187792}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a^{2} + \frac{22135982688518754120688480114644846110965348477570024523770969049268438287200692838447578626968}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} a - \frac{294861129475388379386828415598887399408485370136365576208668524088583703273096787818391208266716}{23276825741052508452506892002830092070151639833618838217544485273920386275987164734676248178283} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 305045817888548.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{85})\), 4.0.614125.2, 4.4.39304000.2, \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-10}, \sqrt{-34})\), 4.0.614125.1, 4.4.39304000.1, \(\Q(\sqrt{-10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-34})\), 4.4.122825.1, 4.0.7860800.1, 4.4.4913.1, 4.0.314432.2, \(\Q(\zeta_{5})\), 4.4.8000.1, 4.0.36125.1, 4.4.2312000.1, 8.0.1544804416000000.25, 8.0.213813760000.1, 8.0.1544804416000000.27, 8.0.377149515625.1, 8.0.1544804416000000.17, 8.8.1544804416000000.3, 8.0.1544804416000000.15, 8.0.61792176640000.40, 8.0.98867482624.1, 8.0.64000000.1, 8.0.5345344000000.1, 8.8.15085980625.1, 8.0.61792176640000.52, 8.0.1305015625.1, 8.8.5345344000000.2, 8.0.61792176640000.6, 8.0.61792176640000.44, 8.0.5345344000000.6, 8.0.5345344000000.4, 16.0.2386420683693101056000000000000.3, 16.0.3818273093908961689600000000.1, 16.0.28572702478336000000000000.2, 16.0.142241757136172119140625.1, 16.0.2386420683693101056000000000000.1, 16.16.2386420683693101056000000000000.1, 16.0.2386420683693101056000000000000.6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed