# Properties

 Label 10.0.216670707.1 Degree $10$ Signature $[0, 5]$ Discriminant $-\,3^{5}\cdot 11^{2}\cdot 7369$ Root discriminant $6.82$ Ramified primes $3, 11, 7369$ Class number $1$ Class group Trivial Galois Group $S_5^2 \wr C_2$ (as 10T43)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

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

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: $$-216670707=-\,3^{5}\cdot 11^{2}\cdot 7369$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.82$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 11, 7369$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $$-a^{6} + a^{5} - a^{4} + a^{3} - a^{2}$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a$$,  $$a^{3} - a^{2}$$,  $$a^{4} - a^{3} + a^{2} - a$$,  $$a^{9} - 2 a^{8} + 3 a^{7} - 3 a^{6} + 3 a^{5} - 2 a^{4} + 2 a^{3} - a^{2} + a - 1$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$1.74293786533$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$S_5^2 \wr C_2$ (as 10T43):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 28800 The 35 conjugacy class representatives for $S_5^2 \wr C_2$ Character table for $S_5^2 \wr C_2$ is not computed

## Intermediate fields

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

## Sibling fields

 Degree 12 sibling: data not computed Degree 20 siblings: data not computed Degree 24 siblings: data not computed Degree 25 sibling: data not computed Degree 30 sibling: data not computed Degree 36 sibling: data not computed Degree 40 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5} 1111.4.2.2x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
7369Data not computed