Properties

Label 8.2.764411904.1
Degree $8$
Signature $[2, 3]$
Discriminant $-764411904$
Root discriminant \(12.89\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 2*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 14*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^8 - 4*x^7 + 2*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 14*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 14, -4, -7, 4, 2, -4, 1]);
 

\( x^{8} - 4x^{7} + 2x^{6} + 4x^{5} - 7x^{4} - 4x^{3} + 14x^{2} - 8x + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(-764411904\) \(\medspace = -\,2^{20}\cdot 3^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(12.89\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(3\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{53}a^{7}+\frac{4}{53}a^{6}-\frac{19}{53}a^{5}+\frac{11}{53}a^{4}-\frac{25}{53}a^{3}+\frac{8}{53}a^{2}+\frac{25}{53}a-\frac{20}{53}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $a-1$, $\frac{101}{53}a^{7}-\frac{338}{53}a^{6}-\frac{11}{53}a^{5}+\frac{369}{53}a^{4}-\frac{458}{53}a^{3}-\frac{676}{53}a^{2}+\frac{935}{53}a-\frac{218}{53}$, $a^{7}-3a^{6}-a^{5}+3a^{4}-4a^{3}-8a^{2}+6a-1$, $\frac{48}{53}a^{7}-\frac{179}{53}a^{6}+\frac{42}{53}a^{5}+\frac{210}{53}a^{4}-\frac{246}{53}a^{3}-\frac{252}{53}a^{2}+\frac{564}{53}a-\frac{112}{53}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 21.9616229603 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 21.9616229603 \cdot 1}{2\sqrt{764411904}}\approx 0.394067220981$

Galois group

Group 1152.157849 (as 8T47):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1152
The 20 conjugacy class representatives for $S_4\wr C_2$
Character table for $S_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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 ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.8.0.1}{8} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.20.30$x^{8} + 2 x^{4} + 8 x^{3} + 28$$4$$2$$20$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
2.1152.4t3.c.a$2$ $ 2^{7} \cdot 3^{2}$ 4.0.4608.1 $D_{4}$ (as 4T3) $1$ $0$
4.373248.6t13.b.a$4$ $ 2^{9} \cdot 3^{6}$ 6.0.1492992.4 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2985984.12t34.b.a$4$ $ 2^{12} \cdot 3^{6}$ 6.0.1492992.4 $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.5971968.12t34.c.a$4$ $ 2^{13} \cdot 3^{6}$ 6.0.1492992.4 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.746496.6t13.b.a$4$ $ 2^{10} \cdot 3^{6}$ 6.0.1492992.4 $C_3^2:D_4$ (as 6T13) $1$ $2$
6.13759414272.12t201.q.a$6$ $ 2^{21} \cdot 3^{8}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $2$
6.382205952.12t202.c.a$6$ $ 2^{19} \cdot 3^{6}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.95551488.8t47.a.a$6$ $ 2^{17} \cdot 3^{6}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $0$
6.13759414272.12t200.e.a$6$ $ 2^{21} \cdot 3^{8}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.178...512.16t1294.a.a$9$ $ 2^{25} \cdot 3^{12}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.713...048.18t272.b.a$9$ $ 2^{27} \cdot 3^{12}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $1$
9.570...384.18t273.c.a$9$ $ 2^{30} \cdot 3^{12}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $1$
9.570...384.18t274.c.a$9$ $ 2^{30} \cdot 3^{12}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.425...464.36t1763.a.a$12$ $ 2^{40} \cdot 3^{18}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.106...616.24t2821.a.a$12$ $ 2^{38} \cdot 3^{18}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $2$
18.146...552.36t1758.a.a$18$ $ 2^{59} \cdot 3^{26}$ 8.2.764411904.1 $S_4\wr C_2$ (as 8T47) $1$ $0$

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 *.