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Properties

Label 22.2.49985887761...0288.1
Degree $22$
Signature $[2, 10]$
Discriminant $2^{22}\cdot 3^{11}\cdot 11^{20}$
Root discriminant $30.64$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{11}^2$ (as 22T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 224, -1720, 7236, -17824, 26464, -22483, 8072, 795, 2080, -5167, 2204, -62, 1360, -2070, 1316, -662, 364, -123, -8, 23, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 23*x^20 - 8*x^19 - 123*x^18 + 364*x^17 - 662*x^16 + 1316*x^15 - 2070*x^14 + 1360*x^13 - 62*x^12 + 2204*x^11 - 5167*x^10 + 2080*x^9 + 795*x^8 + 8072*x^7 - 22483*x^6 + 26464*x^5 - 17824*x^4 + 7236*x^3 - 1720*x^2 + 224*x - 8)
 
gp: K = bnfinit(x^22 - 8*x^21 + 23*x^20 - 8*x^19 - 123*x^18 + 364*x^17 - 662*x^16 + 1316*x^15 - 2070*x^14 + 1360*x^13 - 62*x^12 + 2204*x^11 - 5167*x^10 + 2080*x^9 + 795*x^8 + 8072*x^7 - 22483*x^6 + 26464*x^5 - 17824*x^4 + 7236*x^3 - 1720*x^2 + 224*x - 8, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 23 x^{20} - 8 x^{19} - 123 x^{18} + 364 x^{17} - 662 x^{16} + 1316 x^{15} - 2070 x^{14} + 1360 x^{13} - 62 x^{12} + 2204 x^{11} - 5167 x^{10} + 2080 x^{9} + 795 x^{8} + 8072 x^{7} - 22483 x^{6} + 26464 x^{5} - 17824 x^{4} + 7236 x^{3} - 1720 x^{2} + 224 x - 8 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(499858877615201668877221298700288=2^{22}\cdot 3^{11}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.64$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} + \frac{1}{3} a^{7} - \frac{1}{12} a^{6} - \frac{1}{3} a^{5} + \frac{1}{4} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{7} - \frac{1}{3} a^{6} + \frac{1}{4} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{16} - \frac{1}{36} a^{15} + \frac{1}{36} a^{14} + \frac{1}{36} a^{13} - \frac{1}{36} a^{12} - \frac{1}{12} a^{11} + \frac{1}{36} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{11}{36} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} + \frac{1}{3} a^{4} - \frac{7}{18} a^{3} - \frac{5}{18} a^{2} - \frac{2}{9}$, $\frac{1}{36} a^{17} - \frac{1}{36} a^{14} - \frac{1}{36} a^{12} - \frac{1}{18} a^{11} + \frac{1}{36} a^{10} - \frac{5}{36} a^{8} - \frac{2}{9} a^{7} - \frac{1}{4} a^{6} - \frac{1}{12} a^{5} - \frac{11}{36} a^{4} + \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{36} a^{18} - \frac{1}{36} a^{15} - \frac{1}{36} a^{13} - \frac{1}{18} a^{12} + \frac{1}{36} a^{11} - \frac{5}{36} a^{9} - \frac{2}{9} a^{8} - \frac{1}{4} a^{7} - \frac{1}{12} a^{6} - \frac{11}{36} a^{5} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{4}{9} a$, $\frac{1}{72} a^{19} - \frac{1}{72} a^{17} - \frac{1}{72} a^{16} - \frac{1}{24} a^{15} - \frac{1}{24} a^{14} + \frac{1}{72} a^{13} + \frac{5}{72} a^{12} + \frac{5}{72} a^{11} - \frac{1}{24} a^{10} + \frac{13}{72} a^{9} + \frac{5}{72} a^{8} + \frac{4}{9} a^{7} + \frac{13}{72} a^{6} - \frac{5}{12} a^{5} - \frac{1}{36} a^{4} + \frac{2}{9} a^{3} - \frac{7}{18} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{72} a^{20} - \frac{1}{72} a^{18} - \frac{1}{72} a^{17} - \frac{1}{72} a^{16} + \frac{1}{72} a^{15} - \frac{1}{24} a^{14} + \frac{1}{72} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{7}{72} a^{9} - \frac{1}{18} a^{8} - \frac{5}{24} a^{7} - \frac{1}{12} a^{6} - \frac{13}{36} a^{5} + \frac{11}{36} a^{4} - \frac{5}{18} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{92254294691472494642181768} a^{21} + \frac{21129739948746137328317}{10250477187941388293575752} a^{20} + \frac{154389839476437320015285}{92254294691472494642181768} a^{19} - \frac{77383830781501562990507}{46127147345736247321090884} a^{18} - \frac{15377181689535417715390}{11531786836434061830272721} a^{17} + \frac{50218449803252984042101}{3843928945478020610090907} a^{16} + \frac{1541598390069580654173701}{46127147345736247321090884} a^{15} + \frac{336909068079418981483973}{11531786836434061830272721} a^{14} - \frac{1307785351605259125741575}{46127147345736247321090884} a^{13} + \frac{170059531277981827289146}{11531786836434061830272721} a^{12} - \frac{1064696780071962300379043}{46127147345736247321090884} a^{11} + \frac{287235122213373547585801}{7687857890956041220181814} a^{10} + \frac{18464213644787275932984161}{92254294691472494642181768} a^{9} - \frac{3182703362266327439993365}{92254294691472494642181768} a^{8} - \frac{16908171974888736377243255}{92254294691472494642181768} a^{7} + \frac{15120994637913061067842733}{46127147345736247321090884} a^{6} - \frac{7892425499710478119053967}{46127147345736247321090884} a^{5} + \frac{1428089763083523896007001}{3843928945478020610090907} a^{4} + \frac{321321530436987089417483}{23063573672868123660545442} a^{3} - \frac{1058281136094434015808598}{11531786836434061830272721} a^{2} + \frac{4270721035497921771082637}{11531786836434061830272721} a - \frac{2944968511200116059343665}{11531786836434061830272721}$

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:  $11$
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:  \( 432957757.802 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{11}^2$ (as 22T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 484
The 49 conjugacy class representatives for $D_{11}^2$
Character table for $D_{11}^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \)

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 R R $22$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.11.20.16$x^{11} + 11 x^{10} + 11$$11$$1$$20$$D_{11}$$[2]^{2}$