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)

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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)

Normalized defining 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

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ 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$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$