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Properties

Label 22.14.4098058278...8729.2
Degree $22$
Signature $[14, 4]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 15, 41, -75, -159, 99, -700, 80, 579, -975, 1889, 160, -935, 1546, -1121, -508, 729, -197, -57, 65, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 14 x^{20} + 65 x^{19} - 57 x^{18} - 197 x^{17} + 729 x^{16} - 508 x^{15} - 1121 x^{14} + 1546 x^{13} - 935 x^{12} + 160 x^{11} + 1889 x^{10} - 975 x^{9} + 579 x^{8} + 80 x^{7} - 700 x^{6} + 99 x^{5} - 159 x^{4} - 75 x^{3} + 41 x^{2} + 15 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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:  $23, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47} a^{19} + \frac{17}{47} a^{18} - \frac{3}{47} a^{17} - \frac{19}{47} a^{16} + \frac{1}{47} a^{15} - \frac{11}{47} a^{14} - \frac{22}{47} a^{13} + \frac{15}{47} a^{12} + \frac{7}{47} a^{11} + \frac{5}{47} a^{10} - \frac{20}{47} a^{9} + \frac{15}{47} a^{8} + \frac{1}{47} a^{7} - \frac{5}{47} a^{6} - \frac{22}{47} a^{5} - \frac{8}{47} a^{4} + \frac{12}{47} a^{3} + \frac{21}{47} a^{2} - \frac{17}{47} a + \frac{2}{47}$, $\frac{1}{47} a^{20} - \frac{10}{47} a^{18} - \frac{15}{47} a^{17} - \frac{5}{47} a^{16} + \frac{19}{47} a^{15} - \frac{23}{47} a^{14} + \frac{13}{47} a^{13} - \frac{13}{47} a^{12} - \frac{20}{47} a^{11} - \frac{11}{47} a^{10} - \frac{21}{47} a^{9} - \frac{19}{47} a^{8} - \frac{22}{47} a^{7} + \frac{16}{47} a^{6} - \frac{10}{47} a^{5} + \frac{7}{47} a^{4} + \frac{5}{47} a^{3} + \frac{2}{47} a^{2} + \frac{9}{47} a + \frac{13}{47}$, $\frac{1}{886074417459257567061970228561} a^{21} - \frac{5381274573757033652941917901}{886074417459257567061970228561} a^{20} - \frac{6260106050668971205570604777}{886074417459257567061970228561} a^{19} + \frac{59875204045785342625925181586}{886074417459257567061970228561} a^{18} + \frac{82587176323027901926041406712}{886074417459257567061970228561} a^{17} + \frac{104924798822926122477708004290}{886074417459257567061970228561} a^{16} + \frac{240283569705251450505473279920}{886074417459257567061970228561} a^{15} - \frac{266368216863656062884238082793}{886074417459257567061970228561} a^{14} + \frac{290438208016014419590492209711}{886074417459257567061970228561} a^{13} - \frac{115912143541555708447901723153}{886074417459257567061970228561} a^{12} - \frac{113937858151639036276227180418}{886074417459257567061970228561} a^{11} - \frac{401159460417198546588789886305}{886074417459257567061970228561} a^{10} - \frac{232944096252970807007354410640}{886074417459257567061970228561} a^{9} - \frac{391994599852224898324402138645}{886074417459257567061970228561} a^{8} - \frac{284154250618212195720450567085}{886074417459257567061970228561} a^{7} - \frac{131123306760870282519901545459}{886074417459257567061970228561} a^{6} - \frac{141369091468286219179012245232}{886074417459257567061970228561} a^{5} + \frac{192103742684293964610750510034}{886074417459257567061970228561} a^{4} - \frac{353603888552958954606441716883}{886074417459257567061970228561} a^{3} - \frac{432757859490697004628806310021}{886074417459257567061970228561} a^{2} + \frac{364610717565733767080927488829}{886074417459257567061970228561} a + \frac{384503966897540676073410016723}{886074417459257567061970228561}$

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:  $17$
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:  \( 903801457.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
47Data not computed