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Properties

Label 22.22.9337503149...1168.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{22}\cdot 67^{21}$
Root discriminant $110.69$
Ramified primes $2, 67$
Class number $1$ (GRH)
Class group Trivial (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![-67, 0, 7839, 0, -106195, 0, 512550, 0, -1133506, 0, 1211561, 0, -613452, 0, 160264, 0, -22713, 0, 1742, 0, -67, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 67*x^20 + 1742*x^18 - 22713*x^16 + 160264*x^14 - 613452*x^12 + 1211561*x^10 - 1133506*x^8 + 512550*x^6 - 106195*x^4 + 7839*x^2 - 67)
 
gp: K = bnfinit(x^22 - 67*x^20 + 1742*x^18 - 22713*x^16 + 160264*x^14 - 613452*x^12 + 1211561*x^10 - 1133506*x^8 + 512550*x^6 - 106195*x^4 + 7839*x^2 - 67, 1)
 

Normalized defining polynomial

\( x^{22} - 67 x^{20} + 1742 x^{18} - 22713 x^{16} + 160264 x^{14} - 613452 x^{12} + 1211561 x^{10} - 1133506 x^{8} + 512550 x^{6} - 106195 x^{4} + 7839 x^{2} - 67 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(933750314983937236210361829997141120063111168=2^{22}\cdot 67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.69$
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:  $2, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(268=2^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{268}(1,·)$, $\chi_{268}(3,·)$, $\chi_{268}(129,·)$, $\chi_{268}(9,·)$, $\chi_{268}(267,·)$, $\chi_{268}(81,·)$, $\chi_{268}(75,·)$, $\chi_{268}(259,·)$, $\chi_{268}(139,·)$, $\chi_{268}(149,·)$, $\chi_{268}(89,·)$, $\chi_{268}(25,·)$, $\chi_{268}(27,·)$, $\chi_{268}(225,·)$, $\chi_{268}(243,·)$, $\chi_{268}(119,·)$, $\chi_{268}(193,·)$, $\chi_{268}(43,·)$, $\chi_{268}(241,·)$, $\chi_{268}(179,·)$, $\chi_{268}(265,·)$, $\chi_{268}(187,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29} a^{16} - \frac{1}{29} a^{14} + \frac{13}{29} a^{12} - \frac{4}{29} a^{10} - \frac{11}{29} a^{8} - \frac{11}{29} a^{6} + \frac{5}{29} a^{4} + \frac{7}{29} a^{2} - \frac{5}{29}$, $\frac{1}{29} a^{17} - \frac{1}{29} a^{15} + \frac{13}{29} a^{13} - \frac{4}{29} a^{11} - \frac{11}{29} a^{9} - \frac{11}{29} a^{7} + \frac{5}{29} a^{5} + \frac{7}{29} a^{3} - \frac{5}{29} a$, $\frac{1}{104081} a^{18} + \frac{1722}{104081} a^{16} + \frac{13399}{104081} a^{14} + \frac{42492}{104081} a^{12} - \frac{18677}{104081} a^{10} + \frac{44343}{104081} a^{8} - \frac{21790}{104081} a^{6} - \frac{16753}{104081} a^{4} + \frac{50046}{104081} a^{2} - \frac{7629}{104081}$, $\frac{1}{104081} a^{19} + \frac{1722}{104081} a^{17} + \frac{13399}{104081} a^{15} + \frac{42492}{104081} a^{13} - \frac{18677}{104081} a^{11} + \frac{44343}{104081} a^{9} - \frac{21790}{104081} a^{7} - \frac{16753}{104081} a^{5} + \frac{50046}{104081} a^{3} - \frac{7629}{104081} a$, $\frac{1}{11769933343674148933} a^{20} - \frac{14430902126496}{11769933343674148933} a^{18} - \frac{178006461792639605}{11769933343674148933} a^{16} + \frac{2346707019078143843}{11769933343674148933} a^{14} - \frac{5754079320484931552}{11769933343674148933} a^{12} - \frac{2913900756043386233}{11769933343674148933} a^{10} + \frac{2749579763818108410}{11769933343674148933} a^{8} - \frac{3690627638673721902}{11769933343674148933} a^{6} + \frac{3036278358663837739}{11769933343674148933} a^{4} + \frac{4557385933435253828}{11769933343674148933} a^{2} - \frac{1757122307640656399}{11769933343674148933}$, $\frac{1}{11769933343674148933} a^{21} - \frac{14430902126496}{11769933343674148933} a^{19} - \frac{178006461792639605}{11769933343674148933} a^{17} + \frac{2346707019078143843}{11769933343674148933} a^{15} - \frac{5754079320484931552}{11769933343674148933} a^{13} - \frac{2913900756043386233}{11769933343674148933} a^{11} + \frac{2749579763818108410}{11769933343674148933} a^{9} - \frac{3690627638673721902}{11769933343674148933} a^{7} + \frac{3036278358663837739}{11769933343674148933} a^{5} + \frac{4557385933435253828}{11769933343674148933} a^{3} - \frac{1757122307640656399}{11769933343674148933} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $21$
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:  \( 6014326916364551.0 \) (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

\(\Q(\sqrt{67}) \), 11.11.1822837804551761449.1

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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ $22$

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
2Data not computed
67Data not computed