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Properties

Label 22.8.71980792679...5104.3
Degree $22$
Signature $[8, 7]$
Discriminant $-\,2^{22}\cdot 23^{20}$
Root discriminant $34.59$
Ramified primes $2, 23$
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, 0, 84, 0, 372, 0, -981, 0, -1492, 0, 196, 0, 721, 0, 75, 0, -113, 0, -20, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 5*x^20 - 20*x^18 - 113*x^16 + 75*x^14 + 721*x^12 + 196*x^10 - 1492*x^8 - 981*x^6 + 372*x^4 + 84*x^2 + 1)
 
gp: K = bnfinit(x^22 + 5*x^20 - 20*x^18 - 113*x^16 + 75*x^14 + 721*x^12 + 196*x^10 - 1492*x^8 - 981*x^6 + 372*x^4 + 84*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{22} + 5 x^{20} - 20 x^{18} - 113 x^{16} + 75 x^{14} + 721 x^{12} + 196 x^{10} - 1492 x^{8} - 981 x^{6} + 372 x^{4} + 84 x^{2} + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7198079267989980836471065337135104=-\,2^{22}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.59$
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:  $2, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23302174321} a^{20} - \frac{7840753154}{23302174321} a^{18} - \frac{1277913068}{23302174321} a^{16} + \frac{925076091}{23302174321} a^{14} + \frac{10652755780}{23302174321} a^{12} - \frac{2805522911}{23302174321} a^{10} + \frac{5410715096}{23302174321} a^{8} - \frac{363279063}{23302174321} a^{6} + \frac{9739844632}{23302174321} a^{4} - \frac{98083426}{495790943} a^{2} - \frac{4874936031}{23302174321}$, $\frac{1}{23302174321} a^{21} - \frac{7840753154}{23302174321} a^{19} - \frac{1277913068}{23302174321} a^{17} + \frac{925076091}{23302174321} a^{15} + \frac{10652755780}{23302174321} a^{13} - \frac{2805522911}{23302174321} a^{11} + \frac{5410715096}{23302174321} a^{9} - \frac{363279063}{23302174321} a^{7} + \frac{9739844632}{23302174321} a^{5} - \frac{98083426}{495790943} a^{3} - \frac{4874936031}{23302174321} a$

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:  $14$
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:  \( 260618630.211 \) (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 R $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
2Data not computed
23Data not computed