# Properties

 Label 10.0.2357947691.1 Degree $10$ Signature $[0, 5]$ Discriminant $-\,11^{9}$ Root discriminant $8.65$ Ramified prime $11$ Class number $1$ Class group Trivial Galois Group $C_{10}$ (as 10T1)

# 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]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp: K = bnfinit(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

## Normalizeddefining polynomial

$$x^{10}$$ $$\mathstrut -\mathstrut x^{9}$$ $$\mathstrut +\mathstrut x^{8}$$ $$\mathstrut -\mathstrut x^{7}$$ $$\mathstrut +\mathstrut x^{6}$$ $$\mathstrut -\mathstrut x^{5}$$ $$\mathstrut +\mathstrut x^{4}$$ $$\mathstrut -\mathstrut x^{3}$$ $$\mathstrut +\mathstrut x^{2}$$ $$\mathstrut -\mathstrut x$$ $$\mathstrut +\mathstrut 1$$

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

## Invariants

 Degree: $10$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 5]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$-2357947691=-\,11^{9}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $8.65$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $11$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$11$$ Dirichlet character group: $\lbrace$$\chi_{11}(1,·), \chi_{11}(2,·), \chi_{11}(3,·), \chi_{11}(4,·), \chi_{11}(5,·), \chi_{11}(6,·), \chi_{11}(7,·), \chi_{11}(8,·), \chi_{11}(9,·), \chi_{11}(10,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$

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

## Class group and class number

Trivial group, which has order $1$

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

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $4$ magma: UnitRank(K); sage: UK.rank() gp: K.fu Torsion generator: $$a$$ (order $22$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a - 1$$,  $$a^{2} + 1$$,  $$a^{9} - a^{8} + a^{7}$$,  $$a^{9} + a^{3}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$26.1711060094$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_{10}$ (as 10T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A cyclic group of order 10 The 10 conjugacy class representatives for $C_{10}$ Character table for $C_{10}$

## 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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.11.2t1.1c1$1$ $11$ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.1c1$1$ $11$ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.1c1$1$ $11$ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c2$1$ $11$ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.1c2$1$ $11$ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c3$1$ $11$ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.1c3$1$ $11$ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c4$1$ $11$ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.1c4$1$ $11$ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$

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 *.