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Properties

Label 22.12.1781764468...3423.3
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
Ramified primes $23, 47$
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, -12, 42, 552, -1074, -4458, 3196, 13658, -4342, -16061, 4331, 8067, -666, -3299, 9, 756, 165, -114, -109, 46, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 12*x^20 + 46*x^19 - 109*x^18 - 114*x^17 + 165*x^16 + 756*x^15 + 9*x^14 - 3299*x^13 - 666*x^12 + 8067*x^11 + 4331*x^10 - 16061*x^9 - 4342*x^8 + 13658*x^7 + 3196*x^6 - 4458*x^5 - 1074*x^4 + 552*x^3 + 42*x^2 - 12*x - 1)
 
gp: K = bnfinit(x^22 - 8*x^21 + 12*x^20 + 46*x^19 - 109*x^18 - 114*x^17 + 165*x^16 + 756*x^15 + 9*x^14 - 3299*x^13 - 666*x^12 + 8067*x^11 + 4331*x^10 - 16061*x^9 - 4342*x^8 + 13658*x^7 + 3196*x^6 - 4458*x^5 - 1074*x^4 + 552*x^3 + 42*x^2 - 12*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 12 x^{20} + 46 x^{19} - 109 x^{18} - 114 x^{17} + 165 x^{16} + 756 x^{15} + 9 x^{14} - 3299 x^{13} - 666 x^{12} + 8067 x^{11} + 4331 x^{10} - 16061 x^{9} - 4342 x^{8} + 13658 x^{7} + 3196 x^{6} - 4458 x^{5} - 1074 x^{4} + 552 x^{3} + 42 x^{2} - 12 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47} a^{19} - \frac{10}{47} a^{18} + \frac{22}{47} a^{17} - \frac{8}{47} a^{16} - \frac{12}{47} a^{15} + \frac{14}{47} a^{14} + \frac{9}{47} a^{13} - \frac{9}{47} a^{12} - \frac{5}{47} a^{11} - \frac{6}{47} a^{10} + \frac{10}{47} a^{9} + \frac{9}{47} a^{8} - \frac{15}{47} a^{7} - \frac{19}{47} a^{6} - \frac{21}{47} a^{5} - \frac{17}{47} a^{4} - \frac{16}{47} a^{3} - \frac{19}{47} a^{2} + \frac{7}{47} a - \frac{3}{47}$, $\frac{1}{47} a^{20} + \frac{16}{47} a^{18} - \frac{23}{47} a^{17} + \frac{2}{47} a^{16} - \frac{12}{47} a^{15} + \frac{8}{47} a^{14} - \frac{13}{47} a^{13} - \frac{1}{47} a^{12} - \frac{9}{47} a^{11} - \frac{3}{47} a^{10} + \frac{15}{47} a^{9} - \frac{19}{47} a^{8} + \frac{19}{47} a^{7} - \frac{23}{47} a^{6} + \frac{8}{47} a^{5} + \frac{2}{47} a^{4} + \frac{9}{47} a^{3} + \frac{5}{47} a^{2} + \frac{20}{47} a + \frac{17}{47}$, $\frac{1}{9711071898270111962114145139459} a^{21} + \frac{12409194595077717953795161724}{9711071898270111962114145139459} a^{20} - \frac{102445492111362100539233691779}{9711071898270111962114145139459} a^{19} + \frac{3789979012024775421458135690998}{9711071898270111962114145139459} a^{18} + \frac{548658800226955923303033614468}{9711071898270111962114145139459} a^{17} + \frac{2445772327345159633908457750098}{9711071898270111962114145139459} a^{16} + \frac{240027339767097091850074499327}{9711071898270111962114145139459} a^{15} - \frac{4846834416888365616496036275490}{9711071898270111962114145139459} a^{14} - \frac{2669511263425040413390874988253}{9711071898270111962114145139459} a^{13} + \frac{799863178110626933555986683462}{9711071898270111962114145139459} a^{12} + \frac{4042152624624629896789056711060}{9711071898270111962114145139459} a^{11} - \frac{1004366224527446966023982610768}{9711071898270111962114145139459} a^{10} - \frac{1070951396009082773571098288797}{9711071898270111962114145139459} a^{9} - \frac{1277234637679147040849767214717}{9711071898270111962114145139459} a^{8} + \frac{56697436153753404540233199058}{9711071898270111962114145139459} a^{7} + \frac{627923856297096651663616511879}{9711071898270111962114145139459} a^{6} - \frac{2580440228024436982144645484418}{9711071898270111962114145139459} a^{5} - \frac{3203432267319849599036437741254}{9711071898270111962114145139459} a^{4} - \frac{3217112886403073594100559047505}{9711071898270111962114145139459} a^{3} + \frac{576758795105239083278340700874}{9711071898270111962114145139459} a^{2} - \frac{583957314460705390176370004315}{9711071898270111962114145139459} a - \frac{3665369175660647953742522655697}{9711071898270111962114145139459}$

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:  $16$
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:  \( 106197776.635 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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