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Properties

Label 10.0.532362855424.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 151^{4}$
Root discriminant $14.88$
Ramified primes $2, 151$
Class number $1$
Class group Trivial
Galois group $A_5\times C_2$ (as 10T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -10, 25, -38, 32, -10, 11, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 8*x^7 + 11*x^6 - 10*x^5 + 32*x^4 - 38*x^3 + 25*x^2 - 10*x + 2)
 
gp: K = bnfinit(x^10 - 8*x^7 + 11*x^6 - 10*x^5 + 32*x^4 - 38*x^3 + 25*x^2 - 10*x + 2, 1)
 

Normalized defining polynomial

\( x^{10} - 8 x^{7} + 11 x^{6} - 10 x^{5} + 32 x^{4} - 38 x^{3} + 25 x^{2} - 10 x + 2 \)

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:  \(-532362855424=-\,2^{10}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.88$
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:  $2, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{556} a^{9} + \frac{13}{278} a^{8} - \frac{19}{556} a^{7} - \frac{85}{556} a^{6} + \frac{25}{556} a^{5} + \frac{223}{556} a^{4} - \frac{2}{139} a^{3} + \frac{171}{556} a^{2} + \frac{81}{278} a - \frac{123}{278}$

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:  \( -\frac{155}{278} a^{9} - \frac{137}{556} a^{8} - \frac{87}{556} a^{7} + \frac{1221}{278} a^{6} - \frac{2329}{556} a^{5} + \frac{579}{139} a^{4} - \frac{9057}{556} a^{3} + \frac{8011}{556} a^{2} - \frac{1296}{139} a + \frac{739}{278} \) (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:  \( 334.837297391 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 10T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $A_5\times C_2$
Character table for $A_5\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.364816.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$151$151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.6.4.1$x^{6} + 5285 x^{3} + 39400128$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
3.2e2_151e2.12t76.1c1$3$ $ 2^{2} \cdot 151^{2}$ $x^{10} - 8 x^{7} + 11 x^{6} - 10 x^{5} + 32 x^{4} - 38 x^{3} + 25 x^{2} - 10 x + 2$ $A_5\times C_2$ (as 10T11) $1$ $1$
3.2e4_151e2.12t33.1c1$3$ $ 2^{4} \cdot 151^{2}$ $x^{5} - 2 x^{3} - 2 x^{2} + 5 x - 6$ $A_5$ (as 5T4) $1$ $-1$
3.2e4_151e2.12t33.1c2$3$ $ 2^{4} \cdot 151^{2}$ $x^{5} - 2 x^{3} - 2 x^{2} + 5 x - 6$ $A_5$ (as 5T4) $1$ $-1$
3.2e2_151e2.12t76.1c2$3$ $ 2^{2} \cdot 151^{2}$ $x^{10} - 8 x^{7} + 11 x^{6} - 10 x^{5} + 32 x^{4} - 38 x^{3} + 25 x^{2} - 10 x + 2$ $A_5\times C_2$ (as 10T11) $1$ $1$
* 4.2e4_151e2.5t4.1c1$4$ $ 2^{4} \cdot 151^{2}$ $x^{5} - 2 x^{3} - 2 x^{2} + 5 x - 6$ $A_5$ (as 5T4) $1$ $0$
* 4.2e4_151e2.10t11.1c1$4$ $ 2^{4} \cdot 151^{2}$ $x^{10} - 8 x^{7} + 11 x^{6} - 10 x^{5} + 32 x^{4} - 38 x^{3} + 25 x^{2} - 10 x + 2$ $A_5\times C_2$ (as 10T11) $1$ $0$
5.2e4_151e4.6t12.1c1$5$ $ 2^{4} \cdot 151^{4}$ $x^{5} - 2 x^{3} - 2 x^{2} + 5 x - 6$ $A_5$ (as 5T4) $1$ $1$
5.2e6_151e4.12t75.1c1$5$ $ 2^{6} \cdot 151^{4}$ $x^{10} - 8 x^{7} + 11 x^{6} - 10 x^{5} + 32 x^{4} - 38 x^{3} + 25 x^{2} - 10 x + 2$ $A_5\times C_2$ (as 10T11) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.