# Properties

 Label 22.0.39471584120...9623.1 Degree $22$ Signature $[0, 11]$ Discriminant $-\,23^{21}$ Root discriminant $19.94$ Ramified prime $23$ Class number $3$ Class group $[3]$ Galois Group $C_{22}$ (as 22T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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]);
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)

## Normalizeddefining polynomial

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

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

## Invariants

 Degree: $22$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 11]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$-39471584120695485887249589623=-\,23^{21}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $19.94$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $23$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ 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 $$a$$)

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

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

## Class group and class number

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

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

## Unit group

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

## Galois group

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

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

## Intermediate fields

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

## Frobenius cycle types

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

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