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Properties

Label 22.6.40980582781...8729.2
Degree $22$
Signature $[6, 8]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $2$ (GRH)
Class group $[2]$ (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, -3280, 4336, 3420, 673, -12826, 18311, -37877, 54822, -42755, 17912, -2023, -3027, 3577, -1962, 558, -107, 141, -111, 36, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 111*x^18 + 141*x^17 - 107*x^16 + 558*x^15 - 1962*x^14 + 3577*x^13 - 3027*x^12 - 2023*x^11 + 17912*x^10 - 42755*x^9 + 54822*x^8 - 37877*x^7 + 18311*x^6 - 12826*x^5 + 673*x^4 + 3420*x^3 + 4336*x^2 - 3280*x + 47)
 
gp: K = bnfinit(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 111*x^18 + 141*x^17 - 107*x^16 + 558*x^15 - 1962*x^14 + 3577*x^13 - 3027*x^12 - 2023*x^11 + 17912*x^10 - 42755*x^9 + 54822*x^8 - 37877*x^7 + 18311*x^6 - 12826*x^5 + 673*x^4 + 3420*x^3 + 4336*x^2 - 3280*x + 47, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} + 2 x^{20} + 36 x^{19} - 111 x^{18} + 141 x^{17} - 107 x^{16} + 558 x^{15} - 1962 x^{14} + 3577 x^{13} - 3027 x^{12} - 2023 x^{11} + 17912 x^{10} - 42755 x^{9} + 54822 x^{8} - 37877 x^{7} + 18311 x^{6} - 12826 x^{5} + 673 x^{4} + 3420 x^{3} + 4336 x^{2} - 3280 x + 47 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47} a^{19} - \frac{3}{47} a^{18} - \frac{6}{47} a^{17} + \frac{7}{47} a^{16} - \frac{16}{47} a^{15} - \frac{2}{47} a^{14} - \frac{5}{47} a^{13} - \frac{20}{47} a^{12} - \frac{19}{47} a^{11} - \frac{19}{47} a^{10} + \frac{18}{47} a^{9} - \frac{8}{47} a^{8} + \frac{14}{47} a^{7} + \frac{21}{47} a^{6} - \frac{17}{47} a^{5} - \frac{17}{47} a^{4} + \frac{15}{47} a^{3} - \frac{10}{47} a^{2} - \frac{20}{47} a$, $\frac{1}{47} a^{20} - \frac{15}{47} a^{18} - \frac{11}{47} a^{17} + \frac{5}{47} a^{16} - \frac{3}{47} a^{15} - \frac{11}{47} a^{14} + \frac{12}{47} a^{13} + \frac{15}{47} a^{12} + \frac{18}{47} a^{11} + \frac{8}{47} a^{10} - \frac{1}{47} a^{9} - \frac{10}{47} a^{8} + \frac{16}{47} a^{7} - \frac{1}{47} a^{6} - \frac{21}{47} a^{5} + \frac{11}{47} a^{4} - \frac{12}{47} a^{3} - \frac{3}{47} a^{2} - \frac{13}{47} a$, $\frac{1}{4031272075970082285749472636164185569444769586593} a^{21} - \frac{95820884129888499535340375386065256168807195}{85771746297235793313818566726897565307335523119} a^{20} - \frac{33048158967174714371918383880479768788114635057}{4031272075970082285749472636164185569444769586593} a^{19} - \frac{1409268215336197473475375926891036688390623494026}{4031272075970082285749472636164185569444769586593} a^{18} - \frac{432794410260144138998241877084491822617222805347}{4031272075970082285749472636164185569444769586593} a^{17} + \frac{889069815050176914509271280983235290998033147607}{4031272075970082285749472636164185569444769586593} a^{16} + \frac{750260852123153375115006649477695217500837287571}{4031272075970082285749472636164185569444769586593} a^{15} - \frac{1263486644044421033708533791225284656039242680410}{4031272075970082285749472636164185569444769586593} a^{14} - \frac{691428409030073363803581896944867721738456579798}{4031272075970082285749472636164185569444769586593} a^{13} - \frac{1560895764432060809963033556195423289887319603231}{4031272075970082285749472636164185569444769586593} a^{12} - \frac{589688981512607886277729639439105360148824990623}{4031272075970082285749472636164185569444769586593} a^{11} + \frac{1382898133915827136606051962533857860125026129660}{4031272075970082285749472636164185569444769586593} a^{10} - \frac{83889779738232149895356599684151399607633634344}{4031272075970082285749472636164185569444769586593} a^{9} + \frac{26708083553275757200513513978125274688441006880}{85771746297235793313818566726897565307335523119} a^{8} - \frac{1935343456056636950546368611570071294440508807500}{4031272075970082285749472636164185569444769586593} a^{7} - \frac{1114738717429965680775875538045733009350298400723}{4031272075970082285749472636164185569444769586593} a^{6} + \frac{512935854980100741008020545658979444257584079701}{4031272075970082285749472636164185569444769586593} a^{5} + \frac{1576800715335741146962184360755000353905977204953}{4031272075970082285749472636164185569444769586593} a^{4} + \frac{853506563636256861567424662801235166455643774712}{4031272075970082285749472636164185569444769586593} a^{3} - \frac{207270929141026627046073566044692430592657626151}{4031272075970082285749472636164185569444769586593} a^{2} - \frac{592087662127203022940176703787484316177243303970}{4031272075970082285749472636164185569444769586593} a + \frac{36554865434514162720010502885811945211037941934}{85771746297235793313818566726897565307335523119}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 76447219.2916 \) (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