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Properties

Label 22.0.394...623.1
Degree $22$
Signature $[0, 11]$
Discriminant $-3.947\times 10^{28}$
Root discriminant $19.94$
Ramified prime $23$
Class number $3$
Class group $[3]$
Galois group $C_{22}$ (as 22T1)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
 
gp: K = bnfinit(x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);
 

\(x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)  Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-39471584120695485887249589623\)\(\medspace = -\,23^{21}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.94$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $22$
This field is Galois and abelian over $\Q$.
Conductor:  \(23\)
Dirichlet character group:    $\lbrace$$\chi_{23}(1,·)$, $\chi_{23}(2,·)$, $\chi_{23}(3,·)$, $\chi_{23}(4,·)$, $\chi_{23}(5,·)$, $\chi_{23}(6,·)$, $\chi_{23}(7,·)$, $\chi_{23}(8,·)$, $\chi_{23}(9,·)$, $\chi_{23}(10,·)$, $\chi_{23}(11,·)$, $\chi_{23}(12,·)$, $\chi_{23}(13,·)$, $\chi_{23}(14,·)$, $\chi_{23}(15,·)$, $\chi_{23}(16,·)$, $\chi_{23}(17,·)$, $\chi_{23}(18,·)$, $\chi_{23}(19,·)$, $\chi_{23}(20,·)$, $\chi_{23}(21,·)$, $\chi_{23}(22,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$  Toggle raw display

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

Class group and class number

$C_{3}$, which has order $3$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a \) (order $46$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{4} - a^{3} \),  \( a^{4} - a \),  \( a^{11} + a^{9} + a^{7} \),  \( a^{20} - a^{3} \),  \( a^{16} + a^{6} \),  \( a^{11} + a^{9} \),  \( a^{21} - a^{14} + a^{5} \),  \( a^{11} - a^{8} + a^{5} \),  \( a^{21} - a^{20} - a^{18} - a^{16} + a^{15} + a^{13} + a^{11} - a^{10} - a^{8} + a^{7} - a^{6} + a^{5} + a^{3} - a^{2} - 1 \),  \( a^{16} + a^{8} + 1 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1038656.82438 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 1038656.82438 \cdot 3}{46\sqrt{39471584120695485887249589623}}\approx 0.205433741423$

Galois group

$C_{22}$ (as 22T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/padicField/29.11.0.1}{11} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ $22$ ${\href{/padicField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
23Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.23.2t1.a.a$1$ $ 23 $ \(\Q(\sqrt{-23}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.23.11t1.a.a$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.a$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.b$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.b$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.c$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.c$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.d$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.d$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.e$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.e$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.f$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.f$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.g$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.g$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.h$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.h$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.i$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.i$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.j$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.22t1.a.j$1$ $ 23 $ \(\Q(\zeta_{23})\) $C_{22}$ (as 22T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.