# Properties

 Label 10.0.229345007.1 Degree $10$ Signature $[0, 5]$ Discriminant $-\,47^{5}$ Root discriminant $6.86$ Ramified prime $47$ Class number $1$ Class group Trivial Galois Group $D_5$ (as 10T2)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 6, -3, 11, -3, 11, -3, 6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 6*x^8 - 3*x^7 + 11*x^6 - 3*x^5 + 11*x^4 - 3*x^3 + 6*x^2 - x + 1)
gp: K = bnfinit(x^10 - x^9 + 6*x^8 - 3*x^7 + 11*x^6 - 3*x^5 + 11*x^4 - 3*x^3 + 6*x^2 - x + 1, 1)

## Normalizeddefining polynomial

$$x^{10}$$ $$\mathstrut -\mathstrut x^{9}$$ $$\mathstrut +\mathstrut 6 x^{8}$$ $$\mathstrut -\mathstrut 3 x^{7}$$ $$\mathstrut +\mathstrut 11 x^{6}$$ $$\mathstrut -\mathstrut 3 x^{5}$$ $$\mathstrut +\mathstrut 11 x^{4}$$ $$\mathstrut -\mathstrut 3 x^{3}$$ $$\mathstrut +\mathstrut 6 x^{2}$$ $$\mathstrut -\mathstrut x$$ $$\mathstrut +\mathstrut 1$$

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: $$-229345007=-\,47^{5}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.86$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $47$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$

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

## Class group and class number

Trivial Abelian group, 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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{1}{5} a^{9} + a^{8} + \frac{1}{5} a^{7} + \frac{28}{5} a^{6} - \frac{1}{5} a^{5} + \frac{41}{5} a^{4} - \frac{3}{5} a^{3} + \frac{34}{5} a^{2} - 2 a + \frac{14}{5}$$,  $$\frac{4}{5} a^{9} - 2 a^{8} + \frac{24}{5} a^{7} - \frac{43}{5} a^{6} + \frac{31}{5} a^{5} - \frac{66}{5} a^{4} + \frac{18}{5} a^{3} - \frac{64}{5} a^{2} + 2 a - \frac{19}{5}$$,  $$\frac{3}{5} a^{9} + \frac{18}{5} a^{7} + \frac{4}{5} a^{6} + \frac{37}{5} a^{5} + \frac{8}{5} a^{4} + \frac{31}{5} a^{3} + \frac{2}{5} a^{2} + 2 a - \frac{3}{5}$$,  $$\frac{8}{5} a^{9} - a^{8} + \frac{43}{5} a^{7} - \frac{6}{5} a^{6} + \frac{67}{5} a^{5} + \frac{3}{5} a^{4} + \frac{56}{5} a^{3} - \frac{8}{5} a^{2} + 4 a - \frac{3}{5}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.60186322759$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_5$ (as 10T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 10 The 4 conjugacy class representatives for $D_5$ Character table for $D_5$

## Intermediate fields

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

## Sibling fields

 Degree 5 sibling: 5.1.2209.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2} 47.2.1.2x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2} 47.2.1.2x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$