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Properties

Label 22.0.33934008204...0343.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,7^{11}\cdot 23^{20}$
Root discriminant $45.76$
Ramified primes $7, 23$
Class number $737$ (GRH)
Class group $[737]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2048, -6144, 59904, 1280, 280960, -39424, 528640, -60352, 513224, -57864, 297670, -30697, 110504, -10383, 27309, -2231, 4520, -308, 490, -25, 32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 32*x^20 - 25*x^19 + 490*x^18 - 308*x^17 + 4520*x^16 - 2231*x^15 + 27309*x^14 - 10383*x^13 + 110504*x^12 - 30697*x^11 + 297670*x^10 - 57864*x^9 + 513224*x^8 - 60352*x^7 + 528640*x^6 - 39424*x^5 + 280960*x^4 + 1280*x^3 + 59904*x^2 - 6144*x + 2048)
 
gp: K = bnfinit(x^22 - x^21 + 32*x^20 - 25*x^19 + 490*x^18 - 308*x^17 + 4520*x^16 - 2231*x^15 + 27309*x^14 - 10383*x^13 + 110504*x^12 - 30697*x^11 + 297670*x^10 - 57864*x^9 + 513224*x^8 - 60352*x^7 + 528640*x^6 - 39424*x^5 + 280960*x^4 + 1280*x^3 + 59904*x^2 - 6144*x + 2048, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 32 x^{20} - 25 x^{19} + 490 x^{18} - 308 x^{17} + 4520 x^{16} - 2231 x^{15} + 27309 x^{14} - 10383 x^{13} + 110504 x^{12} - 30697 x^{11} + 297670 x^{10} - 57864 x^{9} + 513224 x^{8} - 60352 x^{7} + 528640 x^{6} - 39424 x^{5} + 280960 x^{4} + 1280 x^{3} + 59904 x^{2} - 6144 x + 2048 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3393400820453274956705986794666660343=-\,7^{11}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.76$
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:  $7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(161=7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{161}(64,·)$, $\chi_{161}(1,·)$, $\chi_{161}(6,·)$, $\chi_{161}(71,·)$, $\chi_{161}(8,·)$, $\chi_{161}(139,·)$, $\chi_{161}(13,·)$, $\chi_{161}(78,·)$, $\chi_{161}(141,·)$, $\chi_{161}(146,·)$, $\chi_{161}(85,·)$, $\chi_{161}(27,·)$, $\chi_{161}(29,·)$, $\chi_{161}(36,·)$, $\chi_{161}(104,·)$, $\chi_{161}(41,·)$, $\chi_{161}(48,·)$, $\chi_{161}(50,·)$, $\chi_{161}(118,·)$, $\chi_{161}(55,·)$, $\chi_{161}(62,·)$, $\chi_{161}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{12} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{7}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} - \frac{1}{32} a^{13} + \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{2} a^{10} + \frac{9}{32} a^{9} + \frac{13}{32} a^{8} - \frac{15}{32} a^{7} - \frac{1}{32} a^{5} + \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{64} a^{17} - \frac{1}{64} a^{16} - \frac{1}{64} a^{14} + \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{1}{4} a^{11} - \frac{23}{64} a^{10} + \frac{13}{64} a^{9} + \frac{17}{64} a^{8} + \frac{31}{64} a^{6} - \frac{9}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{128} a^{18} - \frac{1}{128} a^{17} - \frac{1}{128} a^{15} + \frac{1}{64} a^{14} - \frac{1}{32} a^{13} + \frac{1}{8} a^{12} - \frac{23}{128} a^{11} - \frac{51}{128} a^{10} + \frac{17}{128} a^{9} - \frac{1}{2} a^{8} + \frac{31}{128} a^{7} + \frac{23}{64} a^{6} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4}$, $\frac{1}{256} a^{19} - \frac{1}{256} a^{18} - \frac{1}{256} a^{16} + \frac{1}{128} a^{15} - \frac{1}{64} a^{14} + \frac{1}{16} a^{13} - \frac{23}{256} a^{12} - \frac{51}{256} a^{11} + \frac{17}{256} a^{10} - \frac{1}{4} a^{9} + \frac{31}{256} a^{8} + \frac{23}{128} a^{7} - \frac{15}{32} a^{6} - \frac{1}{16} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{512} a^{20} - \frac{1}{512} a^{19} - \frac{1}{512} a^{17} + \frac{1}{256} a^{16} - \frac{1}{128} a^{15} + \frac{1}{32} a^{14} - \frac{23}{512} a^{13} - \frac{51}{512} a^{12} + \frac{17}{512} a^{11} + \frac{3}{8} a^{10} - \frac{225}{512} a^{9} - \frac{105}{256} a^{8} - \frac{15}{64} a^{7} - \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4198948279583180568167395694894981940224} a^{21} + \frac{682575004155511121604674000566093311}{4198948279583180568167395694894981940224} a^{20} + \frac{1529010520821239381211552745948909693}{1049737069895795142041848923723745485056} a^{19} - \frac{3385569081459357273490426551713722373}{4198948279583180568167395694894981940224} a^{18} + \frac{13194927402780068075080935034258355185}{2099474139791590284083697847447490970112} a^{17} + \frac{5177477043514911668560003464342742379}{524868534947897571020924461861872742528} a^{16} - \frac{4317292125367848581752216429683073755}{524868534947897571020924461861872742528} a^{15} + \frac{99072824708671230129619904066669268713}{4198948279583180568167395694894981940224} a^{14} + \frac{343457921996397553239957965680987737037}{4198948279583180568167395694894981940224} a^{13} - \frac{249326560402649988377869478261997633051}{4198948279583180568167395694894981940224} a^{12} + \frac{378416152537549155957751158065065476645}{1049737069895795142041848923723745485056} a^{11} + \frac{159392549411834162947338641730336954835}{4198948279583180568167395694894981940224} a^{10} + \frac{974874580542341375681702787267357987175}{2099474139791590284083697847447490970112} a^{9} - \frac{170399137094854449605240286313130625807}{1049737069895795142041848923723745485056} a^{8} - \frac{68063624376158298323356871563984772313}{524868534947897571020924461861872742528} a^{7} - \frac{12103283376931042615236798936912673397}{262434267473948785510462230930936371264} a^{6} - \frac{772023714608635965450956535648597905}{16402141717121799094403889433183523204} a^{5} + \frac{6545150034073904022443744636382211439}{16402141717121799094403889433183523204} a^{4} + \frac{2258446334731245556583412849811160541}{32804283434243598188807778866367046408} a^{3} + \frac{2986699171454319812630971250092200183}{8201070858560899547201944716591761602} a^{2} + \frac{2889021512734964302840225649934734921}{8201070858560899547201944716591761602} a - \frac{1430264706273562298983405762770693249}{4100535429280449773600972358295880801}$

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

Class group and class number

$C_{737}$, which has order $737$ (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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 1038656.82438 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
7Data not computed
$23$23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$