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Properties

Label 8.4.1584788925456.2
Degree $8$
Signature $[4, 2]$
Discriminant $2^{4}\cdot 3^{4}\cdot 11^{4}\cdot 17^{4}$
Root discriminant $33.50$
Ramified primes $2, 3, 11, 17$
Class number $2$
Class group $[2]$
Galois group $(A_4\wr C_2):C_2$ (as 8T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 0, 50, -25, 0, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 4*x^6 - 25*x^4 + 50*x^3 + 16)
 
gp: K = bnfinit(x^8 - 4*x^7 + 4*x^6 - 25*x^4 + 50*x^3 + 16, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16 \)

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:  \(1584788925456=2^{4}\cdot 3^{4}\cdot 11^{4}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.50$
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:  $2, 3, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1000} a^{7} + \frac{3}{50} a^{6} - \frac{39}{250} a^{5} + \frac{2}{125} a^{4} - \frac{1}{1000} a^{3} - \frac{7}{500} a^{2} + \frac{13}{125} a - \frac{43}{125}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( 2057.82996063 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(A_4\wr C_2):C_2$ (as 8T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$
Character table for $(A_4\wr C_2):C_2$

Intermediate fields

\(\Q(\sqrt{561}) \)

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 siblings: 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 R ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
1.3_17.2t1.1c1$1$ $ 3 \cdot 17 $ $x^{2} - x + 13$ $C_2$ (as 2T1) $1$ $-1$
* 1.3_11_17.2t1.1c1$1$ $ 3 \cdot 11 \cdot 17 $ $x^{2} - x - 140$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3_17.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 17 $ $x^{3} - x^{2} + x - 3$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e2_11_17e2.6t3.1c1$2$ $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 17^{2}$ $x^{6} - x^{5} - 11 x^{4} + 127 x^{3} + 418 x^{2} - 2094 x - 21722$ $D_{6}$ (as 6T3) $1$ $0$
2.2e2_11.3t2.1c1$2$ $ 2^{2} \cdot 11 $ $x^{3} - x^{2} + x + 1$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3_11e2_17.6t3.1c1$2$ $ 2^{2} \cdot 3 \cdot 11^{2} \cdot 17 $ $x^{6} - 2 x^{5} - 9 x^{4} + 82 x^{3} - 47 x^{2} - 360 x - 3753$ $D_{6}$ (as 6T3) $1$ $0$
4.2e2_3e2_11e2_17e2.6t9.1c1$4$ $ 2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 17^{2}$ $x^{6} - 2 x^{5} - x^{4} - 15 x^{3} + 18 x^{2} + 17 x - 68$ $S_3^2$ (as 6T9) $1$ $0$
* 6.2e4_3e3_11e3_17e3.8t45.1c1$6$ $ 2^{4} \cdot 3^{3} \cdot 11^{3} \cdot 17^{3}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $2$
6.2e4_3e3_11e3_17e3.12t161.1c1$6$ $ 2^{4} \cdot 3^{3} \cdot 11^{3} \cdot 17^{3}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-2$
9.2e12_3e6_11e3_17e6.18t185.1c1$9$ $ 2^{12} \cdot 3^{6} \cdot 11^{3} \cdot 17^{6}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.2e12_3e3_11e3_17e3.12t165.1c1$9$ $ 2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 17^{3}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
9.2e12_3e3_11e6_17e3.18t185.1c1$9$ $ 2^{12} \cdot 3^{3} \cdot 11^{6} \cdot 17^{3}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.2e12_3e6_11e6_17e6.18t179.1c1$9$ $ 2^{12} \cdot 3^{6} \cdot 11^{6} \cdot 17^{6}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
12.2e14_3e6_11e6_17e6.24t1504.1c1$12$ $ 2^{14} \cdot 3^{6} \cdot 11^{6} \cdot 17^{6}$ $x^{8} - 4 x^{7} + 4 x^{6} - 25 x^{4} + 50 x^{3} + 16$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $0$

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 *.