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Properties

Label 22.16.8719272932...7207.1
Degree $22$
Signature $[16, 3]$
Discriminant $-\,23^{21}\cdot 47^{2}$
Root discriminant $28.30$
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, 66, -267, -372, 2749, -1577, -8172, 11912, 5227, -21040, 9530, 11338, -13613, 2314, 4443, -3255, 450, 513, -297, 40, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 18*x^20 + 40*x^19 - 297*x^18 + 513*x^17 + 450*x^16 - 3255*x^15 + 4443*x^14 + 2314*x^13 - 13613*x^12 + 11338*x^11 + 9530*x^10 - 21040*x^9 + 5227*x^8 + 11912*x^7 - 8172*x^6 - 1577*x^5 + 2749*x^4 - 372*x^3 - 267*x^2 + 66*x + 1)
 
gp: K = bnfinit(x^22 - 8*x^21 + 18*x^20 + 40*x^19 - 297*x^18 + 513*x^17 + 450*x^16 - 3255*x^15 + 4443*x^14 + 2314*x^13 - 13613*x^12 + 11338*x^11 + 9530*x^10 - 21040*x^9 + 5227*x^8 + 11912*x^7 - 8172*x^6 - 1577*x^5 + 2749*x^4 - 372*x^3 - 267*x^2 + 66*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 18 x^{20} + 40 x^{19} - 297 x^{18} + 513 x^{17} + 450 x^{16} - 3255 x^{15} + 4443 x^{14} + 2314 x^{13} - 13613 x^{12} + 11338 x^{11} + 9530 x^{10} - 21040 x^{9} + 5227 x^{8} + 11912 x^{7} - 8172 x^{6} - 1577 x^{5} + 2749 x^{4} - 372 x^{3} - 267 x^{2} + 66 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-87192729322616328324934343477207=-\,23^{21}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{375643737801459618923} a^{21} + \frac{95764531244339678091}{375643737801459618923} a^{20} + \frac{134804221956919192380}{375643737801459618923} a^{19} + \frac{22951288782361839397}{375643737801459618923} a^{18} - \frac{156705797405584962348}{375643737801459618923} a^{17} - \frac{125010396713980309811}{375643737801459618923} a^{16} - \frac{59831270061577267752}{375643737801459618923} a^{15} + \frac{175622233066067988644}{375643737801459618923} a^{14} + \frac{120962805641933359520}{375643737801459618923} a^{13} + \frac{133444278569583871619}{375643737801459618923} a^{12} + \frac{92566126925132138023}{375643737801459618923} a^{11} + \frac{45947449127058966474}{375643737801459618923} a^{10} - \frac{108324147956598303550}{375643737801459618923} a^{9} - \frac{96111441638309554266}{375643737801459618923} a^{8} + \frac{62865881947971437877}{375643737801459618923} a^{7} - \frac{116195096846455042230}{375643737801459618923} a^{6} - \frac{130850518952208080151}{375643737801459618923} a^{5} - \frac{50197467837550042287}{375643737801459618923} a^{4} + \frac{135929908796452180236}{375643737801459618923} a^{3} - \frac{61644995993961680314}{375643737801459618923} a^{2} + \frac{169577059678403644458}{375643737801459618923} a + \frac{94567398576867623566}{375643737801459618923}$

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:  $18$
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:  \( 177605096.839 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ 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
47Data not computed