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Properties

Label 22.22.6006316524...7952.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{33}\cdot 3^{11}\cdot 23^{21}$
Root discriminant $97.71$
Ramified primes $2, 3, 23$
Class number $2$ (GRH)
Class group $[2]$ (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![-8344332288, 0, 30595885056, 0, -33145542144, 0, 16572771072, 0, -4603547520, 0, 781208064, 0, -85131648, 0, 6080832, 0, -283176, 0, 8280, 0, -138, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 138*x^20 + 8280*x^18 - 283176*x^16 + 6080832*x^14 - 85131648*x^12 + 781208064*x^10 - 4603547520*x^8 + 16572771072*x^6 - 33145542144*x^4 + 30595885056*x^2 - 8344332288)
 
gp: K = bnfinit(x^22 - 138*x^20 + 8280*x^18 - 283176*x^16 + 6080832*x^14 - 85131648*x^12 + 781208064*x^10 - 4603547520*x^8 + 16572771072*x^6 - 33145542144*x^4 + 30595885056*x^2 - 8344332288, 1)
 

Normalized defining polynomial

\( x^{22} - 138 x^{20} + 8280 x^{18} - 283176 x^{16} + 6080832 x^{14} - 85131648 x^{12} + 781208064 x^{10} - 4603547520 x^{8} + 16572771072 x^{6} - 33145542144 x^{4} + 30595885056 x^{2} - 8344332288 \)

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:  \(60063165247472201954266758470414053725437952=2^{33}\cdot 3^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.71$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(245,·)$, $\chi_{552}(5,·)$, $\chi_{552}(193,·)$, $\chi_{552}(265,·)$, $\chi_{552}(149,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(361,·)$, $\chi_{552}(413,·)$, $\chi_{552}(389,·)$, $\chi_{552}(289,·)$, $\chi_{552}(293,·)$, $\chi_{552}(169,·)$, $\chi_{552}(365,·)$, $\chi_{552}(221,·)$, $\chi_{552}(49,·)$, $\chi_{552}(53,·)$, $\chi_{552}(73,·)$, $\chi_{552}(121,·)$, $\chi_{552}(125,·)$, $\chi_{552}(341,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{279936} a^{14}$, $\frac{1}{279936} a^{15}$, $\frac{1}{1679616} a^{16}$, $\frac{1}{1679616} a^{17}$, $\frac{1}{10077696} a^{18}$, $\frac{1}{10077696} a^{19}$, $\frac{1}{60466176} a^{20}$, $\frac{1}{60466176} a^{21}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 226492733704000 \) (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

\(\Q(\sqrt{138}) \), \(\Q(\zeta_{23})^+\)

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 R R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
3Data not computed
23Data not computed