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Properties

Label 22.6.40980582781...8729.1
Degree $22$
Signature $[6, 8]$
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![47, -806, 6786, -12927, 2232, -2646, 10213, -5866, -2690, -5015, 1174, 712, 806, 390, -22, 160, -114, -1, -22, -1, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - x^19 - 22*x^18 - x^17 - 114*x^16 + 160*x^15 - 22*x^14 + 390*x^13 + 806*x^12 + 712*x^11 + 1174*x^10 - 5015*x^9 - 2690*x^8 - 5866*x^7 + 10213*x^6 - 2646*x^5 + 2232*x^4 - 12927*x^3 + 6786*x^2 - 806*x + 47)
 
gp: K = bnfinit(x^22 - x^21 + x^20 - x^19 - 22*x^18 - x^17 - 114*x^16 + 160*x^15 - 22*x^14 + 390*x^13 + 806*x^12 + 712*x^11 + 1174*x^10 - 5015*x^9 - 2690*x^8 - 5866*x^7 + 10213*x^6 - 2646*x^5 + 2232*x^4 - 12927*x^3 + 6786*x^2 - 806*x + 47, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + x^{20} - x^{19} - 22 x^{18} - x^{17} - 114 x^{16} + 160 x^{15} - 22 x^{14} + 390 x^{13} + 806 x^{12} + 712 x^{11} + 1174 x^{10} - 5015 x^{9} - 2690 x^{8} - 5866 x^{7} + 10213 x^{6} - 2646 x^{5} + 2232 x^{4} - 12927 x^{3} + 6786 x^{2} - 806 x + 47 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{174883771709751143444715484039468850714144116723} a^{21} - \frac{80484626567360129110536382634530168171470499338}{174883771709751143444715484039468850714144116723} a^{20} - \frac{13936375699903299955303913864654219605379723829}{174883771709751143444715484039468850714144116723} a^{19} - \frac{60571177708071907254313158173980865768940525305}{174883771709751143444715484039468850714144116723} a^{18} + \frac{36432263072581920619113643484717848523887298499}{174883771709751143444715484039468850714144116723} a^{17} + \frac{41320223803545446647775749778005378216601021676}{174883771709751143444715484039468850714144116723} a^{16} + \frac{55124790841368215120375425809509274614219546240}{174883771709751143444715484039468850714144116723} a^{15} + \frac{59068625785658521358965097224182769283393059022}{174883771709751143444715484039468850714144116723} a^{14} + \frac{16877371784930962740937239376728541044585824825}{174883771709751143444715484039468850714144116723} a^{13} - \frac{82654432288391711337510775377094552785418555775}{174883771709751143444715484039468850714144116723} a^{12} - \frac{74016565232891959453180646637716815095337942112}{174883771709751143444715484039468850714144116723} a^{11} - \frac{25615336717913259018082070184977670074610651577}{174883771709751143444715484039468850714144116723} a^{10} - \frac{61120153802278562073069253604750628468444049381}{174883771709751143444715484039468850714144116723} a^{9} + \frac{11135947205019267224864902997672775286582147546}{174883771709751143444715484039468850714144116723} a^{8} + \frac{35931197316843576328207083950312283535627819628}{174883771709751143444715484039468850714144116723} a^{7} - \frac{29559424476632235846694345150222142898999192557}{174883771709751143444715484039468850714144116723} a^{6} + \frac{35082737813082507899463062863818848314623515668}{174883771709751143444715484039468850714144116723} a^{5} - \frac{54078463926544072732393102091025695543442842120}{174883771709751143444715484039468850714144116723} a^{4} - \frac{17881245598493886287590403250956851663854912486}{174883771709751143444715484039468850714144116723} a^{3} + \frac{37775117335820607902184604294027803824779235212}{174883771709751143444715484039468850714144116723} a^{2} + \frac{76507036735428209122529503314888666583954004713}{174883771709751143444715484039468850714144116723} a + \frac{49090279070681853869045433642960103527837030017}{174883771709751143444715484039468850714144116723}$

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:  $13$
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:  \( 123704206.751 \) (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