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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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content 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)
 
Copy content gp:K = bnfinit(y^22 - y^21 + y^20 - y^19 + y^18 - y^17 + y^16 - y^15 + y^14 - y^13 + y^12 - y^11 + y^10 - y^9 + y^8 - y^7 + y^6 - y^5 + y^4 - y^3 + y^2 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); 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);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(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)
 

\( x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $22$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 11]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-39471584120695485887249589623\) \(\medspace = -\,23^{21}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(19.94\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $23^{21/22}\approx 19.944865695037844$
Ramified primes:   \(23\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-23}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{22}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
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.
Reflex fields:  unavailable$^{1024}$

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

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

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{3}$, which has order $3$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $3$

Unit group

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

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1038656.82438 \cdot 3}{46\cdot\sqrt{39471584120695485887249589623}}\cr\approx \mathstrut & 0.205433741423 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula 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) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula 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); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); 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); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(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); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$

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.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(23\) Copy content Toggle raw display 23.1.22.21a1.1$x^{22} + 23$$22$$1$$21$22T1$$[\ ]_{22}$$

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)