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Properties

Label 8.4.1479200000.1
Degree $8$
Signature $[4, 2]$
Discriminant $2^{8}\cdot 5^{5}\cdot 43^{2}$
Root discriminant $14.00$
Ramified primes $2, 5, 43$
Class number $1$
Class group Trivial
Galois group $A_4^2:C_4$ (as 8T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 19, 0, 1, -10, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^6 - 10*x^5 + x^4 + 19*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^8 - x^6 - 10*x^5 + x^4 + 19*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1479200000=2^{8}\cdot 5^{5}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.00$
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:  $2, 5, 43$
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 \(a\))

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

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:  $5$
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:  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:  \( 47.8891829444 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4^2:C_4$ (as 8T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 13 conjugacy class representatives for $A_4^2:C_4$
Character table for $A_4^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \)

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 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.2e2_5_43.4t1.1c1$1$ $ 2^{2} \cdot 5 \cdot 43 $ $x^{4} + 215 x^{2} + 9245$ $C_4$ (as 4T1) $0$ $-1$
1.2e2_5_43.4t1.1c2$1$ $ 2^{2} \cdot 5 \cdot 43 $ $x^{4} + 215 x^{2} + 9245$ $C_4$ (as 4T1) $0$ $-1$
4.2e4_5e3_43e2.6t10.2c1$4$ $ 2^{4} \cdot 5^{3} \cdot 43^{2}$ $x^{6} - 2 x^{5} + 4 x^{4} - 6 x^{3} + 4 x^{2} - 12 x - 9$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.2e4_5e3_43e2.6t10.1c1$4$ $ 2^{4} \cdot 5^{3} \cdot 43^{2}$ $x^{6} - 2 x^{5} + 4 x^{4} - 6 x^{3} + 4 x^{2} - 12 x - 9$ $C_3^2:C_4$ (as 6T10) $1$ $0$
6.2e12_5e5_43e4.12t160.2c1$6$ $ 2^{12} \cdot 5^{5} \cdot 43^{4}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $1$ $-2$
* 6.2e8_5e4_43e2.8t46.1c1$6$ $ 2^{8} \cdot 5^{4} \cdot 43^{2}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $1$ $2$
9.2e16_5e6_43e4.16t1030.2c1$9$ $ 2^{16} \cdot 5^{6} \cdot 43^{4}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $1$ $1$
9.2e16_5e7_43e4.18t184.2c1$9$ $ 2^{16} \cdot 5^{7} \cdot 43^{4}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $1$ $1$
9.2e18_5e7_43e5.36t766.2c1$9$ $ 2^{18} \cdot 5^{7} \cdot 43^{5}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $0$ $-1$
9.2e18_5e7_43e5.36t766.2c2$9$ $ 2^{18} \cdot 5^{7} \cdot 43^{5}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $0$ $-1$
12.2e20_5e9_43e6.24t1506.1c1$12$ $ 2^{20} \cdot 5^{9} \cdot 43^{6}$ $x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1$ $A_4^2:C_4$ (as 8T46) $1$ $0$

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 *.