Properties

Label 10.0.364067099.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,11^{3}\cdot 523^{2}$
Root discriminant $7.18$
Ramified primes $11, 523$
Class number $1$
Class group Trivial
Galois Group $C_2 \wr S_5$ (as 10T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 3, -2, 2, 1, -2, 0, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 2*x^6 + x^5 + 2*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)
gp: K = bnfinit(x^10 - x^9 - 2*x^6 + x^5 + 2*x^4 - 2*x^3 + 3*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{10} \) \(\mathstrut -\mathstrut x^{9} \) \(\mathstrut -\mathstrut 2 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-364067099=-\,11^{3}\cdot 523^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $7.18$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $11, 523$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{21} a^{9} + \frac{1}{21} a^{8} + \frac{2}{21} a^{7} + \frac{4}{21} a^{6} + \frac{2}{7} a^{5} - \frac{8}{21} a^{4} + \frac{1}{3} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{10}{21}$

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:  \( \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{9}{7} a^{4} + a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{6}{7} \),  \( a \),  \( \frac{4}{21} a^{9} + \frac{4}{21} a^{8} + \frac{8}{21} a^{7} - \frac{5}{21} a^{6} - \frac{6}{7} a^{5} - \frac{11}{21} a^{4} - \frac{2}{3} a^{3} + \frac{2}{7} a^{2} + \frac{8}{7} a - \frac{2}{21} \),  \( \frac{8}{21} a^{9} + \frac{8}{21} a^{8} - \frac{5}{21} a^{7} - \frac{10}{21} a^{6} - \frac{5}{7} a^{5} - \frac{22}{21} a^{4} + \frac{2}{3} a^{3} + \frac{4}{7} a^{2} + \frac{2}{7} a + \frac{17}{21} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.826936802139 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2 \wr S_5$ (as 10T39):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 3840
The 36 conjugacy class representatives for $C_2 \wr S_5$
Character table for $C_2 \wr S_5$ is not computed

Intermediate fields

5.1.5753.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
523Data not computed