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Properties

Label 10.0.26214400000000.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{26}\cdot 5^{8}$
Root discriminant $21.97$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $(C_5^2 : C_4) : C_2$ (as 10T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26, -96, 118, -24, -58, 40, 2, -16, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 13*x^8 - 16*x^7 + 2*x^6 + 40*x^5 - 58*x^4 - 24*x^3 + 118*x^2 - 96*x + 26)
 
gp: K = bnfinit(x^10 - 4*x^9 + 13*x^8 - 16*x^7 + 2*x^6 + 40*x^5 - 58*x^4 - 24*x^3 + 118*x^2 - 96*x + 26, 1)
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} + 13 x^{8} - 16 x^{7} + 2 x^{6} + 40 x^{5} - 58 x^{4} - 24 x^{3} + 118 x^{2} - 96 x + 26 \)

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:  \(-26214400000000=-\,2^{26}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{25} a^{8} + \frac{2}{25} a^{7} + \frac{1}{25} a^{6} - \frac{8}{25} a^{5} + \frac{1}{5} a^{4} + \frac{12}{25} a^{3} - \frac{6}{25} a^{2} + \frac{2}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{7} + \frac{6}{25} a^{5} + \frac{2}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{25} a^{2} - \frac{2}{5} a - \frac{3}{25}$

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:  \( \frac{3}{5} a^{9} - 2 a^{8} + \frac{32}{5} a^{7} - \frac{26}{5} a^{6} - \frac{14}{5} a^{5} + \frac{111}{5} a^{4} - 20 a^{3} - \frac{146}{5} a^{2} + \frac{258}{5} a - \frac{103}{5} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3285.52322575 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5:F_5$ (as 10T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $(C_5^2 : C_4) : C_2$
Character table for $(C_5^2 : C_4) : C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \)

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 25 sibling: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.24.5$x^{8} - 15$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.5.6.4$x^{5} + 20 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$