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Properties

Label 22.22.6577635578...8125.1
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 1297^{10}$
Root discriminant $58.14$
Ramified primes $5, 1297$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{22}$ (as 22T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -5, 133, 486, -4409, -4986, 40785, 4085, -122742, 13148, 158022, -11106, -100878, 1614, 35042, 676, -6909, -243, 766, 27, -44, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 44*x^20 + 27*x^19 + 766*x^18 - 243*x^17 - 6909*x^16 + 676*x^15 + 35042*x^14 + 1614*x^13 - 100878*x^12 - 11106*x^11 + 158022*x^10 + 13148*x^9 - 122742*x^8 + 4085*x^7 + 40785*x^6 - 4986*x^5 - 4409*x^4 + 486*x^3 + 133*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 44*x^20 + 27*x^19 + 766*x^18 - 243*x^17 - 6909*x^16 + 676*x^15 + 35042*x^14 + 1614*x^13 - 100878*x^12 - 11106*x^11 + 158022*x^10 + 13148*x^9 - 122742*x^8 + 4085*x^7 + 40785*x^6 - 4986*x^5 - 4409*x^4 + 486*x^3 + 133*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 44 x^{20} + 27 x^{19} + 766 x^{18} - 243 x^{17} - 6909 x^{16} + 676 x^{15} + 35042 x^{14} + 1614 x^{13} - 100878 x^{12} - 11106 x^{11} + 158022 x^{10} + 13148 x^{9} - 122742 x^{8} + 4085 x^{7} + 40785 x^{6} - 4986 x^{5} - 4409 x^{4} + 486 x^{3} + 133 x^{2} - 5 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(657763557836310291639991612795361328125=5^{11}\cdot 1297^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.14$
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:  $5, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{17} - \frac{2}{5} a^{16} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{17} + \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{275} a^{20} + \frac{1}{25} a^{19} - \frac{3}{55} a^{18} + \frac{89}{275} a^{17} - \frac{31}{275} a^{16} + \frac{63}{275} a^{15} + \frac{2}{55} a^{14} - \frac{28}{275} a^{13} - \frac{94}{275} a^{12} - \frac{19}{55} a^{11} + \frac{119}{275} a^{10} - \frac{58}{275} a^{9} + \frac{89}{275} a^{8} - \frac{2}{11} a^{7} - \frac{14}{275} a^{6} - \frac{118}{275} a^{5} - \frac{109}{275} a^{4} - \frac{1}{11} a^{3} - \frac{27}{275} a^{2} - \frac{68}{275} a + \frac{63}{275}$, $\frac{1}{23063418380320308821533423325} a^{21} - \frac{14350953681378145495158624}{23063418380320308821533423325} a^{20} + \frac{377932964412613520357117382}{4612683676064061764306684665} a^{19} - \frac{98478749752540363113741661}{2096674398210937165593947575} a^{18} - \frac{334791559601604563145598136}{23063418380320308821533423325} a^{17} - \frac{2406009566902017973422848737}{23063418380320308821533423325} a^{16} - \frac{1933699014815123866085586813}{4612683676064061764306684665} a^{15} + \frac{3462355564993981822778956907}{23063418380320308821533423325} a^{14} + \frac{5510763053749300710823572396}{23063418380320308821533423325} a^{13} - \frac{390520135113253112368939279}{4612683676064061764306684665} a^{12} + \frac{391002356955820888813877539}{23063418380320308821533423325} a^{11} + \frac{1848968327736383008461256722}{23063418380320308821533423325} a^{10} - \frac{4876702728321583671972071686}{23063418380320308821533423325} a^{9} - \frac{234438890901946725045224174}{922536735212812352861336933} a^{8} - \frac{1166199197129699975261891684}{23063418380320308821533423325} a^{7} - \frac{6857352385489047092674760523}{23063418380320308821533423325} a^{6} - \frac{10004634971992546682453300524}{23063418380320308821533423325} a^{5} + \frac{123605721560365451900335208}{419334879642187433118789515} a^{4} - \frac{1616761819959036682957080762}{23063418380320308821533423325} a^{3} + \frac{7009404270377498298548731717}{23063418380320308821533423325} a^{2} + \frac{10464843206615052504632077983}{23063418380320308821533423325} a - \frac{139961370102238925237533144}{4612683676064061764306684665}$

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:  $21$
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:  \( 3460009379450 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{22}$ (as 22T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 44
The 14 conjugacy class representatives for $D_{22}$
Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{5}) \), 11.11.3670285774226257.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 22 sibling: 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 $22$ $22$ R $22$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ $22$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1297Data not computed