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Properties

Label 22.12.1781764468...3423.4
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
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![-137, 1340, -4987, 8564, -5851, -2016, 10061, -18488, 15063, 9384, -23809, 6022, 12319, -7084, -3117, 3136, 91, -586, 30, 78, -10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 - 10*x^20 + 78*x^19 + 30*x^18 - 586*x^17 + 91*x^16 + 3136*x^15 - 3117*x^14 - 7084*x^13 + 12319*x^12 + 6022*x^11 - 23809*x^10 + 9384*x^9 + 15063*x^8 - 18488*x^7 + 10061*x^6 - 2016*x^5 - 5851*x^4 + 8564*x^3 - 4987*x^2 + 1340*x - 137)
 
gp: K = bnfinit(x^22 - 5*x^21 - 10*x^20 + 78*x^19 + 30*x^18 - 586*x^17 + 91*x^16 + 3136*x^15 - 3117*x^14 - 7084*x^13 + 12319*x^12 + 6022*x^11 - 23809*x^10 + 9384*x^9 + 15063*x^8 - 18488*x^7 + 10061*x^6 - 2016*x^5 - 5851*x^4 + 8564*x^3 - 4987*x^2 + 1340*x - 137, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} - 10 x^{20} + 78 x^{19} + 30 x^{18} - 586 x^{17} + 91 x^{16} + 3136 x^{15} - 3117 x^{14} - 7084 x^{13} + 12319 x^{12} + 6022 x^{11} - 23809 x^{10} + 9384 x^{9} + 15063 x^{8} - 18488 x^{7} + 10061 x^{6} - 2016 x^{5} - 5851 x^{4} + 8564 x^{3} - 4987 x^{2} + 1340 x - 137 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{139} a^{20} - \frac{67}{139} a^{19} - \frac{31}{139} a^{18} - \frac{28}{139} a^{17} - \frac{25}{139} a^{16} - \frac{8}{139} a^{15} + \frac{17}{139} a^{14} + \frac{37}{139} a^{13} + \frac{64}{139} a^{12} + \frac{22}{139} a^{11} - \frac{68}{139} a^{10} - \frac{19}{139} a^{9} - \frac{51}{139} a^{8} - \frac{8}{139} a^{7} - \frac{32}{139} a^{6} - \frac{62}{139} a^{5} + \frac{26}{139} a^{4} + \frac{18}{139} a^{3} - \frac{8}{139} a^{2} - \frac{65}{139} a + \frac{56}{139}$, $\frac{1}{152004242062668703917068744611081} a^{21} - \frac{124841826094726151867076917867}{152004242062668703917068744611081} a^{20} - \frac{39688519454028612470679518717676}{152004242062668703917068744611081} a^{19} - \frac{51673382650302704482541561734319}{152004242062668703917068744611081} a^{18} - \frac{69300125051285026648945442248890}{152004242062668703917068744611081} a^{17} + \frac{57282236714204534337276626878433}{152004242062668703917068744611081} a^{16} + \frac{32047215652331282671416359980416}{152004242062668703917068744611081} a^{15} - \frac{27870256561003045888419627889246}{152004242062668703917068744611081} a^{14} + \frac{60769083586784319147581456938781}{152004242062668703917068744611081} a^{13} - \frac{21200225098601868799023263602893}{152004242062668703917068744611081} a^{12} - \frac{12733103454517021251237394655533}{152004242062668703917068744611081} a^{11} - \frac{55753027643971675680005775080926}{152004242062668703917068744611081} a^{10} + \frac{7054786143397951473326243824311}{152004242062668703917068744611081} a^{9} - \frac{40983934457309180323353748477160}{152004242062668703917068744611081} a^{8} - \frac{20427808234691057634268623750665}{152004242062668703917068744611081} a^{7} - \frac{33213588940581456765119894542846}{152004242062668703917068744611081} a^{6} + \frac{38531090589241059473636783028047}{152004242062668703917068744611081} a^{5} + \frac{31338289768648764888487859519748}{152004242062668703917068744611081} a^{4} + \frac{19847132932978002892904932879761}{152004242062668703917068744611081} a^{3} + \frac{10877547887161319154212671560929}{152004242062668703917068744611081} a^{2} - \frac{67638793080782311825842809989716}{152004242062668703917068744611081} a - \frac{60327279457577698714268069507380}{152004242062668703917068744611081}$

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:  $16$
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:  \( 102536895.286 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ 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
23Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$