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Properties

Label 22.14.3790988231...9009.2
Degree $22$
Signature $[14, 4]$
Discriminant $23^{20}\cdot 47^{2}$
Root discriminant $24.54$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T23

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -10, 29, 22, -50, -4, -40, -36, 30, 27, 115, 27, 30, -36, -40, -4, -50, 22, 29, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 10*x^20 + 29*x^19 + 22*x^18 - 50*x^17 - 4*x^16 - 40*x^15 - 36*x^14 + 30*x^13 + 27*x^12 + 115*x^11 + 27*x^10 + 30*x^9 - 36*x^8 - 40*x^7 - 4*x^6 - 50*x^5 + 22*x^4 + 29*x^3 - 10*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^22 - 3*x^21 - 10*x^20 + 29*x^19 + 22*x^18 - 50*x^17 - 4*x^16 - 40*x^15 - 36*x^14 + 30*x^13 + 27*x^12 + 115*x^11 + 27*x^10 + 30*x^9 - 36*x^8 - 40*x^7 - 4*x^6 - 50*x^5 + 22*x^4 + 29*x^3 - 10*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 10 x^{20} + 29 x^{19} + 22 x^{18} - 50 x^{17} - 4 x^{16} - 40 x^{15} - 36 x^{14} + 30 x^{13} + 27 x^{12} + 115 x^{11} + 27 x^{10} + 30 x^{9} - 36 x^{8} - 40 x^{7} - 4 x^{6} - 50 x^{5} + 22 x^{4} + 29 x^{3} - 10 x^{2} - 3 x + 1 \)

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:  \(3790988231418101231518884499009=23^{20}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.54$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{8}{47} a^{17} - \frac{20}{47} a^{16} + \frac{7}{47} a^{15} - \frac{8}{47} a^{14} - \frac{17}{47} a^{13} - \frac{4}{47} a^{12} + \frac{17}{47} a^{11} - \frac{16}{47} a^{10} + \frac{2}{47} a^{9} - \frac{16}{47} a^{8} + \frac{17}{47} a^{7} - \frac{4}{47} a^{6} - \frac{17}{47} a^{5} - \frac{8}{47} a^{4} + \frac{7}{47} a^{3} - \frac{20}{47} a^{2} - \frac{8}{47} a + \frac{1}{47}$, $\frac{1}{47} a^{19} + \frac{10}{47} a^{17} - \frac{12}{47} a^{16} + \frac{1}{47} a^{15} + \frac{13}{47} a^{14} + \frac{1}{47} a^{13} - \frac{15}{47} a^{12} - \frac{21}{47} a^{11} + \frac{15}{47} a^{10} - \frac{17}{47} a^{8} - \frac{9}{47} a^{7} - \frac{2}{47} a^{6} - \frac{3}{47} a^{5} - \frac{10}{47} a^{4} - \frac{11}{47} a^{3} + \frac{20}{47} a^{2} - \frac{16}{47} a + \frac{8}{47}$, $\frac{1}{73789013} a^{20} - \frac{26139}{73789013} a^{19} + \frac{228028}{73789013} a^{18} - \frac{28332978}{73789013} a^{17} - \frac{1509928}{73789013} a^{16} - \frac{5720546}{73789013} a^{15} - \frac{3249885}{73789013} a^{14} + \frac{29260693}{73789013} a^{13} + \frac{764421}{1569979} a^{12} - \frac{9912110}{73789013} a^{11} - \frac{6995968}{73789013} a^{10} - \frac{30321837}{73789013} a^{9} + \frac{34357808}{73789013} a^{8} + \frac{21410798}{73789013} a^{7} + \frac{21869779}{73789013} a^{6} - \frac{8860504}{73789013} a^{5} + \frac{60051}{73789013} a^{4} - \frac{7923251}{73789013} a^{3} + \frac{15927818}{73789013} a^{2} + \frac{23523546}{73789013} a + \frac{6279917}{73789013}$, $\frac{1}{73789013} a^{21} - \frac{78428}{73789013} a^{19} + \frac{710252}{73789013} a^{18} - \frac{5158011}{73789013} a^{17} - \frac{562472}{1569979} a^{16} + \frac{6327992}{73789013} a^{15} - \frac{33858750}{73789013} a^{14} - \frac{32996413}{73789013} a^{13} - \frac{28235874}{73789013} a^{12} - \frac{16994741}{73789013} a^{11} + \frac{21893880}{73789013} a^{10} - \frac{36870146}{73789013} a^{9} - \frac{1622903}{73789013} a^{8} + \frac{3184907}{73789013} a^{7} + \frac{3379045}{73789013} a^{6} + \frac{218054}{73789013} a^{5} - \frac{23928952}{73789013} a^{4} - \frac{280097}{73789013} a^{3} + \frac{9873}{28153} a^{2} + \frac{2833503}{73789013} a + \frac{18865887}{73789013}$

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:  \( 24067678.7128 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T23:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 11264
The 104 conjugacy class representatives for t22n23 are not computed
Character table for t22n23 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
$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}$
47Data not computed