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Properties

Label 22.16.1781764468...3423.3
Degree $22$
Signature $[16, 3]$
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![-1, 1, 86, 85, -1230, 273, 6225, -9343, -1218, 14512, -11283, -4782, 11203, -2772, -4041, 2224, 616, -607, -25, 93, -11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 - 11*x^20 + 93*x^19 - 25*x^18 - 607*x^17 + 616*x^16 + 2224*x^15 - 4041*x^14 - 2772*x^13 + 11203*x^12 - 4782*x^11 - 11283*x^10 + 14512*x^9 - 1218*x^8 - 9343*x^7 + 6225*x^6 + 273*x^5 - 1230*x^4 + 85*x^3 + 86*x^2 + x - 1)
 
gp: K = bnfinit(x^22 - 5*x^21 - 11*x^20 + 93*x^19 - 25*x^18 - 607*x^17 + 616*x^16 + 2224*x^15 - 4041*x^14 - 2772*x^13 + 11203*x^12 - 4782*x^11 - 11283*x^10 + 14512*x^9 - 1218*x^8 - 9343*x^7 + 6225*x^6 + 273*x^5 - 1230*x^4 + 85*x^3 + 86*x^2 + x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} - 11 x^{20} + 93 x^{19} - 25 x^{18} - 607 x^{17} + 616 x^{16} + 2224 x^{15} - 4041 x^{14} - 2772 x^{13} + 11203 x^{12} - 4782 x^{11} - 11283 x^{10} + 14512 x^{9} - 1218 x^{8} - 9343 x^{7} + 6225 x^{6} + 273 x^{5} - 1230 x^{4} + 85 x^{3} + 86 x^{2} + x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 3]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{12457443373063050850991} a^{21} + \frac{5220876777715281011545}{12457443373063050850991} a^{20} - \frac{5923550984258736859667}{12457443373063050850991} a^{19} + \frac{4129958098181340229464}{12457443373063050850991} a^{18} - \frac{4538681119477797049814}{12457443373063050850991} a^{17} + \frac{4115512379786764917007}{12457443373063050850991} a^{16} + \frac{6006540386844578269204}{12457443373063050850991} a^{15} - \frac{1557641598586730948643}{12457443373063050850991} a^{14} + \frac{4255468952783677116598}{12457443373063050850991} a^{13} + \frac{2711606515793735763356}{12457443373063050850991} a^{12} - \frac{2335458462160339781986}{12457443373063050850991} a^{11} + \frac{1616262724704714800821}{12457443373063050850991} a^{10} + \frac{5837663066819391030604}{12457443373063050850991} a^{9} - \frac{1231038128234335447977}{12457443373063050850991} a^{8} - \frac{48325913715575415515}{12457443373063050850991} a^{7} - \frac{6076482190568765026649}{12457443373063050850991} a^{6} - \frac{3888965959314546415767}{12457443373063050850991} a^{5} + \frac{613252159676558372174}{12457443373063050850991} a^{4} + \frac{2381964062398530217399}{12457443373063050850991} a^{3} + \frac{2102249033503935379842}{12457443373063050850991} a^{2} - \frac{1847023254686097062126}{12457443373063050850991} a - \frac{6188436135022238636089}{12457443373063050850991}$

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:  $18$
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:  \( 262731486.109 \) (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
47Data not computed