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Properties

Label 22.0.54621986108...3839.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,23^{20}\cdot 47\cdot 6771937$
Root discriminant $42.11$
Ramified primes $23, 47, 6771937$
Class number $372$ (GRH)
Class group $[372]$ (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2048, -1024, 6144, -2816, 9728, -4096, 10496, -4048, 8472, -2972, 5336, -1683, 2668, -743, 1059, -253, 328, -64, 76, -11, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 12*x^20 - 11*x^19 + 76*x^18 - 64*x^17 + 328*x^16 - 253*x^15 + 1059*x^14 - 743*x^13 + 2668*x^12 - 1683*x^11 + 5336*x^10 - 2972*x^9 + 8472*x^8 - 4048*x^7 + 10496*x^6 - 4096*x^5 + 9728*x^4 - 2816*x^3 + 6144*x^2 - 1024*x + 2048)
 
gp: K = bnfinit(x^22 - x^21 + 12*x^20 - 11*x^19 + 76*x^18 - 64*x^17 + 328*x^16 - 253*x^15 + 1059*x^14 - 743*x^13 + 2668*x^12 - 1683*x^11 + 5336*x^10 - 2972*x^9 + 8472*x^8 - 4048*x^7 + 10496*x^6 - 4096*x^5 + 9728*x^4 - 2816*x^3 + 6144*x^2 - 1024*x + 2048, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 12 x^{20} - 11 x^{19} + 76 x^{18} - 64 x^{17} + 328 x^{16} - 253 x^{15} + 1059 x^{14} - 743 x^{13} + 2668 x^{12} - 1683 x^{11} + 5336 x^{10} - 2972 x^{9} + 8472 x^{8} - 4048 x^{7} + 10496 x^{6} - 4096 x^{5} + 9728 x^{4} - 2816 x^{3} + 6144 x^{2} - 1024 x + 2048 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-546219861083080897861027662501393839=-\,23^{20}\cdot 47\cdot 6771937\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.11$
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:  $23, 47, 6771937$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{3}{16} a^{8} + \frac{3}{16} a^{7} + \frac{5}{16} a^{6} - \frac{1}{2} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{1}{32} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{3}{32} a^{9} + \frac{3}{32} a^{8} - \frac{11}{32} a^{7} + \frac{1}{4} a^{6} + \frac{1}{32} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{17} - \frac{1}{64} a^{16} + \frac{1}{64} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{3}{64} a^{10} + \frac{3}{64} a^{9} - \frac{11}{64} a^{8} + \frac{1}{8} a^{7} + \frac{1}{64} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{18} - \frac{1}{128} a^{17} + \frac{1}{128} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{3}{128} a^{11} - \frac{61}{128} a^{10} + \frac{53}{128} a^{9} - \frac{7}{16} a^{8} + \frac{1}{128} a^{7} + \frac{1}{16} a^{6} - \frac{7}{16} a^{5} + \frac{7}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{256} a^{19} - \frac{1}{256} a^{18} + \frac{1}{256} a^{16} - \frac{1}{64} a^{15} + \frac{1}{64} a^{14} - \frac{1}{32} a^{13} + \frac{3}{256} a^{12} - \frac{61}{256} a^{11} + \frac{53}{256} a^{10} - \frac{7}{32} a^{9} + \frac{1}{256} a^{8} - \frac{15}{32} a^{7} + \frac{9}{32} a^{6} - \frac{9}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{512} a^{20} - \frac{1}{512} a^{19} + \frac{1}{512} a^{17} - \frac{1}{128} a^{16} + \frac{1}{128} a^{15} - \frac{1}{64} a^{14} + \frac{3}{512} a^{13} - \frac{61}{512} a^{12} + \frac{53}{512} a^{11} - \frac{7}{64} a^{10} + \frac{1}{512} a^{9} + \frac{17}{64} a^{8} - \frac{23}{64} a^{7} - \frac{9}{64} a^{6} - \frac{1}{2} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a$, $\frac{1}{1024} a^{21} - \frac{1}{1024} a^{20} + \frac{1}{1024} a^{18} - \frac{1}{256} a^{17} + \frac{1}{256} a^{16} - \frac{1}{128} a^{15} + \frac{3}{1024} a^{14} - \frac{61}{1024} a^{13} + \frac{53}{1024} a^{12} + \frac{57}{128} a^{11} - \frac{511}{1024} a^{10} - \frac{47}{128} a^{9} - \frac{23}{128} a^{8} - \frac{9}{128} a^{7} - \frac{1}{4} a^{6} - \frac{1}{32} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$

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

Class group and class number

$C_{372}$, which has order $372$ (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:  $10$
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:  \( 1038656.82438 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ 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
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
6771937Data not computed