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Properties

Label 22.16.565...896.2
Degree $22$
Signature $[16, 3]$
Discriminant $-5.650\times 10^{37}$
Root discriminant \(52.00\)
Ramified primes $2,1297$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{22}$ (as 22T32)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1)
 
Copy content gp:K = bnfinit(y^22 - 17*y^20 + 74*y^18 + 62*y^16 - 919*y^14 + 1316*y^12 + 196*y^10 - 1389*y^8 + 887*y^6 - 194*y^4 + 7*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1)
 

\( x^{22} - 17 x^{20} + 74 x^{18} + 62 x^{16} - 919 x^{14} + 1316 x^{12} + 196 x^{10} - 1389 x^{8} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $22$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[16, 3]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-56501459388151144478039723653407440896\) \(\medspace = -\,2^{22}\cdot 1297^{10}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(52.00\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(1297\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{14}+\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{11275}a^{18}-\frac{129}{11275}a^{16}+\frac{79}{451}a^{14}-\frac{27}{451}a^{12}-\frac{2119}{11275}a^{10}+\frac{3569}{11275}a^{8}-\frac{104}{275}a^{6}-\frac{4589}{11275}a^{4}-\frac{3262}{11275}a^{2}+\frac{1108}{11275}$, $\frac{1}{11275}a^{19}-\frac{129}{11275}a^{17}+\frac{79}{451}a^{15}-\frac{27}{451}a^{13}-\frac{2119}{11275}a^{11}+\frac{3569}{11275}a^{9}-\frac{104}{275}a^{7}-\frac{4589}{11275}a^{5}-\frac{3262}{11275}a^{3}+\frac{1108}{11275}a$, $\frac{1}{1228975}a^{20}-\frac{21}{1228975}a^{18}+\frac{6703}{111725}a^{16}+\frac{107469}{245795}a^{14}-\frac{383954}{1228975}a^{12}+\frac{295622}{1228975}a^{10}-\frac{24712}{1228975}a^{8}-\frac{314016}{1228975}a^{6}+\frac{430186}{1228975}a^{4}+\frac{577872}{1228975}a^{2}+\frac{532329}{1228975}$, $\frac{1}{1228975}a^{21}-\frac{21}{1228975}a^{19}+\frac{6703}{111725}a^{17}+\frac{107469}{245795}a^{15}-\frac{383954}{1228975}a^{13}+\frac{295622}{1228975}a^{11}-\frac{24712}{1228975}a^{9}-\frac{314016}{1228975}a^{7}+\frac{430186}{1228975}a^{5}+\frac{577872}{1228975}a^{3}+\frac{532329}{1228975}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $18$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{6511009}{1228975}a^{21}-\frac{107391114}{1228975}a^{19}+\frac{427321022}{1228975}a^{17}+\frac{3034876}{5995}a^{15}-\frac{5677333511}{1228975}a^{13}+\frac{5682756898}{1228975}a^{11}+\frac{4266724542}{1228975}a^{9}-\frac{7005363144}{1228975}a^{7}+\frac{2153430049}{1228975}a^{5}-\frac{29683127}{1228975}a^{3}-\frac{21564314}{1228975}a$, $\frac{1089754}{49159}a^{21}-\frac{40974776}{111725}a^{19}+\frac{1810574074}{1228975}a^{17}+\frac{45706218}{22345}a^{15}-\frac{4777487884}{245795}a^{13}+\frac{24966240294}{1228975}a^{11}+\frac{1516224681}{111725}a^{9}-\frac{30197945601}{1228975}a^{7}+\frac{948621784}{111725}a^{5}-\frac{575133223}{1228975}a^{3}-\frac{58877833}{1228975}a$, $\frac{1089754}{49159}a^{21}-\frac{40974776}{111725}a^{19}+\frac{1810574074}{1228975}a^{17}+\frac{45706218}{22345}a^{15}-\frac{4777487884}{245795}a^{13}+\frac{24966240294}{1228975}a^{11}+\frac{1516224681}{111725}a^{9}-\frac{30197945601}{1228975}a^{7}+\frac{948621784}{111725}a^{5}-\frac{575133223}{1228975}a^{3}-\frac{60106808}{1228975}a$, $\frac{249966}{111725}a^{20}-\frac{45349064}{1228975}a^{18}+\frac{36084764}{245795}a^{16}+\frac{1281833}{5995}a^{14}-\frac{2396756189}{1228975}a^{12}+\frac{2398237944}{1228975}a^{10}+\frac{1796901921}{1228975}a^{8}-\frac{2948548204}{1228975}a^{6}+\frac{911872823}{1228975}a^{4}-\frac{21783827}{1228975}a^{2}-\frac{213409}{49159}$, $\frac{740617}{29975}a^{20}-\frac{502251272}{1228975}a^{18}+\frac{2016133391}{1228975}a^{16}+\frac{561943883}{245795}a^{14}-\frac{26615124438}{1228975}a^{12}+\frac{2520388459}{111725}a^{10}+\frac{18710024171}{1228975}a^{8}-\frac{819253432}{29975}a^{6}+\frac{11483131782}{1228975}a^{4}-\frac{596645671}{1228975}a^{2}-\frac{65607217}{1228975}$, $\frac{6928117}{1228975}a^{21}-\frac{114199718}{1228975}a^{19}+\frac{90690019}{245795}a^{17}+\frac{133572357}{245795}a^{15}-\frac{6038721378}{1228975}a^{13}+\frac{5976603513}{1228975}a^{11}+\frac{4663300337}{1228975}a^{9}-\frac{7454934208}{1228975}a^{7}+\frac{2159325301}{1228975}a^{5}+\frac{53816516}{1228975}a^{3}-\frac{685626}{22345}a$, $\frac{898643}{1228975}a^{21}-\frac{2863671}{245795}a^{19}+\frac{50676512}{1228975}a^{17}+\frac{23772338}{245795}a^{15}-\frac{735512287}{1228975}a^{13}+\frac{347096879}{1228975}a^{11}+\frac{1027184321}{1228975}a^{9}-\frac{11858959}{22345}a^{7}-\frac{231428359}{1228975}a^{5}+\frac{35391497}{245795}a^{3}-\frac{17624934}{1228975}a$, $\frac{6511009}{1228975}a^{21}-\frac{107391114}{1228975}a^{19}+\frac{427321022}{1228975}a^{17}+\frac{3034876}{5995}a^{15}-\frac{5677333511}{1228975}a^{13}+\frac{5682756898}{1228975}a^{11}+\frac{4266724542}{1228975}a^{9}-\frac{7005363144}{1228975}a^{7}+\frac{2153430049}{1228975}a^{5}-\frac{29683127}{1228975}a^{3}-\frac{20335339}{1228975}a$, $\frac{3034508}{245795}a^{20}-\frac{50163978}{245795}a^{18}+\frac{18275402}{22345}a^{16}+\frac{282486899}{245795}a^{14}-\frac{2656684499}{245795}a^{12}+\frac{2747382967}{245795}a^{10}+\frac{1888132794}{245795}a^{8}-\frac{3335800706}{245795}a^{6}+\frac{224881860}{49159}a^{4}-\frac{54273722}{245795}a^{2}-\frac{1401481}{49159}$, $\frac{4885649}{1228975}a^{21}+\frac{45753}{49159}a^{20}-\frac{81244766}{1228975}a^{19}-\frac{18636597}{1228975}a^{18}+\frac{1616588}{5995}a^{17}+\frac{71373043}{1228975}a^{16}+\frac{85205481}{245795}a^{15}+\frac{24594538}{245795}a^{14}-\frac{4332625871}{1228975}a^{13}-\frac{194116644}{245795}a^{12}+\frac{4819995756}{1228975}a^{11}+\frac{73591898}{111725}a^{10}+\frac{2755218234}{1228975}a^{9}+\frac{880897782}{1228975}a^{8}-\frac{523321881}{111725}a^{7}-\frac{1046150772}{1228975}a^{6}+\frac{2182163507}{1228975}a^{5}+\frac{208590153}{1228975}a^{4}-\frac{148198528}{1228975}a^{3}+\frac{25057559}{1228975}a^{2}-\frac{653933}{49159}a-\frac{2507196}{1228975}$, $\frac{1089754}{49159}a^{21}+\frac{2805743}{245795}a^{20}-\frac{40974776}{111725}a^{19}-\frac{232183293}{1228975}a^{18}+\frac{1810574074}{1228975}a^{17}+\frac{933774067}{1228975}a^{16}+\frac{45706218}{22345}a^{15}+\frac{257892361}{245795}a^{14}-\frac{4777487884}{245795}a^{13}-\frac{44773961}{4469}a^{12}+\frac{24966240294}{1228975}a^{11}+\frac{12927403957}{1228975}a^{10}+\frac{1516224681}{111725}a^{9}+\frac{8559766188}{1228975}a^{8}-\frac{30197945601}{1228975}a^{7}-\frac{15632852758}{1228975}a^{6}+\frac{948621784}{111725}a^{5}+\frac{5413456347}{1228975}a^{4}-\frac{575133223}{1228975}a^{3}-\frac{296426169}{1228975}a^{2}-\frac{58877833}{1228975}a-\frac{31300854}{1228975}$, $\frac{4100088}{1228975}a^{21}+\frac{1811267}{1228975}a^{20}-\frac{67729026}{1228975}a^{19}-\frac{30137486}{1228975}a^{18}+\frac{270888416}{1228975}a^{17}+\frac{123117167}{1228975}a^{16}+\frac{76694642}{245795}a^{15}+\frac{31457112}{245795}a^{14}-\frac{3579565802}{1228975}a^{13}-\frac{146319848}{111725}a^{12}+\frac{3681090318}{1228975}a^{11}+\frac{359526206}{245795}a^{10}+\frac{2520279587}{1228975}a^{9}+\frac{205925154}{245795}a^{8}-\frac{4442806066}{1228975}a^{7}-\frac{2151407156}{1228975}a^{6}+\frac{310652202}{245795}a^{5}+\frac{799617378}{1228975}a^{4}-\frac{100828528}{1228975}a^{3}-\frac{50547868}{1228975}a^{2}-\frac{969477}{111725}a-\frac{4885649}{1228975}$, $\frac{8590941}{1228975}a^{21}-\frac{6473061}{1228975}a^{20}-\frac{142075442}{1228975}a^{19}+\frac{107233164}{1228975}a^{18}+\frac{570037262}{1228975}a^{17}-\frac{17297263}{49159}a^{16}+\frac{31863210}{49159}a^{15}-\frac{117931852}{245795}a^{14}-\frac{7529063979}{1228975}a^{13}+\frac{5695299739}{1228975}a^{12}+\frac{7824277961}{1228975}a^{11}-\frac{6041796679}{1228975}a^{10}+\frac{5327101644}{1228975}a^{9}-\frac{3924440066}{1228975}a^{8}-\frac{864556562}{111725}a^{7}+\frac{7307428294}{1228975}a^{6}+\frac{128550336}{49159}a^{5}-\frac{2535507578}{1228975}a^{4}-\frac{135414486}{1228975}a^{3}+\frac{130759617}{1228975}a^{2}-\frac{22227199}{1228975}a+\frac{2934188}{245795}$, $\frac{65522668}{1228975}a^{21}-\frac{34127741}{1228975}a^{20}-\frac{1084618047}{1228975}a^{19}+\frac{112990062}{245795}a^{18}+\frac{174568825}{49159}a^{17}-\frac{206677214}{111725}a^{16}+\frac{240462241}{49159}a^{15}-\frac{626078047}{245795}a^{14}-\frac{57529224242}{1228975}a^{13}+\frac{29969035009}{1228975}a^{12}+\frac{60532582167}{1228975}a^{11}-\frac{31542141568}{1228975}a^{10}+\frac{39868455548}{1228975}a^{9}-\frac{20782208552}{1228975}a^{8}-\frac{73192802277}{1228975}a^{7}+\frac{1526108808}{49159}a^{6}+\frac{25437813459}{1228975}a^{5}-\frac{13231258207}{1228975}a^{4}-\frac{1363514806}{1228975}a^{3}+\frac{138764388}{245795}a^{2}-\frac{5982821}{49159}a+\frac{76254723}{1228975}$, $\frac{2127618}{1228975}a^{21}-\frac{2772472}{245795}a^{20}-\frac{7042186}{245795}a^{19}+\frac{45885728}{245795}a^{18}+\frac{141620662}{1228975}a^{17}-\frac{184540992}{245795}a^{16}+\frac{39011628}{245795}a^{15}-\frac{254768526}{245795}a^{14}-\frac{1864940312}{1228975}a^{13}+\frac{2433051161}{245795}a^{12}+\frac{1964427979}{1228975}a^{11}-\frac{2555302622}{245795}a^{10}+\frac{1268063421}{1228975}a^{9}-\frac{1687747412}{245795}a^{8}-\frac{42896164}{22345}a^{7}+\frac{56141693}{4469}a^{6}+\frac{858672466}{1228975}a^{5}-\frac{1074674344}{245795}a^{4}-\frac{12292733}{245795}a^{3}+\frac{60080806}{245795}a^{2}-\frac{9022109}{1228975}a+\frac{6392863}{245795}$, $\frac{26218144}{1228975}a^{21}+\frac{10378162}{1228975}a^{20}-\frac{86820158}{245795}a^{19}-\frac{171780011}{1228975}a^{18}+\frac{42633391}{29975}a^{17}+\frac{691077797}{1228975}a^{16}+\frac{95964425}{49159}a^{15}+\frac{190472652}{245795}a^{14}-\frac{23030981161}{1228975}a^{13}-\frac{9108623558}{1228975}a^{12}+\frac{24307514702}{1228975}a^{11}+\frac{1916495943}{245795}a^{10}+\frac{15883784368}{1228975}a^{9}+\frac{1258878482}{245795}a^{8}-\frac{5872054614}{245795}a^{7}-\frac{11576013156}{1228975}a^{6}+\frac{10264124743}{1228975}a^{5}+\frac{4051051018}{1228975}a^{4}-\frac{114391071}{245795}a^{3}-\frac{224981318}{1228975}a^{2}-\frac{59676262}{1228975}a-\frac{23489759}{1228975}$, $\frac{30997941}{1228975}a^{21}+\frac{44227657}{1228975}a^{20}-\frac{512497331}{1228975}a^{19}-\frac{731679912}{1228975}a^{18}+\frac{2054685993}{1228975}a^{17}+\frac{2938838351}{1228975}a^{16}+\frac{115209645}{49159}a^{15}+\frac{816712409}{245795}a^{14}-\frac{2467286689}{111725}a^{13}-\frac{3525489113}{111725}a^{12}+\frac{28128581302}{1228975}a^{11}+\frac{40500786159}{1228975}a^{10}+\frac{19170470503}{1228975}a^{9}+\frac{27148761231}{1228975}a^{8}-\frac{34063008361}{1228975}a^{7}-\frac{49028356287}{1228975}a^{6}+\frac{11610276171}{1228975}a^{5}+\frac{16856503502}{1228975}a^{4}-\frac{644304218}{1228975}a^{3}-\frac{21879261}{29975}a^{2}-\frac{57202661}{1228975}a-\frac{97571532}{1228975}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 239801436281 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{3}\cdot 239801436281 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.259305355985085 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 17*x^20 + 74*x^18 + 62*x^16 - 919*x^14 + 1316*x^12 + 196*x^10 - 1389*x^8 + 887*x^6 - 194*x^4 + 7*x^2 + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^{10}.D_{22}$ (as 22T32):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 45056
The 200 conjugacy class representatives for $C_2^{10}.D_{22}$
Character table for $C_2^{10}.D_{22}$

Intermediate fields

11.11.3670285774226257.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.12.56501459388151144478039723653407440896.13

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $22$ ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{9}$ $22$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ $22$ $22$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{7}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.11.2.22a88.2$x^{22} + 2 x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{17} + 2 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{8} + 2 x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 2 x^{3} + 4 x^{2} + 7$$2$$11$$22$not computednot computed
\(1297\) Copy content Toggle raw display $\Q_{1297}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1297}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)