Properties

Label 8.4.32788343808.3
Degree $8$
Signature $[4, 2]$
Discriminant $32788343808$
Root discriminant \(20.63\)
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 + 3*x^6 - 14*x^5 - 18*x^4 + 12*x^3 + 31*x^2 + 18*x + 3)
 
gp: K = bnfinit(x^8 + 3*x^6 - 14*x^5 - 18*x^4 + 12*x^3 + 31*x^2 + 18*x + 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 18, 31, 12, -18, -14, 3, 0, 1]);
 

\( x^{8} + 3x^{6} - 14x^{5} - 18x^{4} + 12x^{3} + 31x^{2} + 18x + 3 \) 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:  $[4, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(32788343808\) \(\medspace = 2^{10}\cdot 3^{7}\cdot 11^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(20.63\)
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:   \(2\), \(3\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{3}a^{3}-\frac{1}{2}$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{4}a-\frac{1}{4}$ 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:  $5$
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:   $\frac{1}{12}a^{7}+\frac{1}{4}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{13}{2}a^{3}+\frac{13}{6}a^{2}+\frac{43}{4}a+\frac{15}{4}$, $\frac{5}{6}a^{7}-\frac{1}{2}a^{6}+3a^{5}-\frac{40}{3}a^{4}-6a^{3}+12a^{2}+\frac{31}{2}a+\frac{7}{2}$, $\frac{5}{6}a^{7}+\frac{8}{3}a^{5}-\frac{34}{3}a^{4}-14a^{3}+\frac{28}{3}a^{2}+\frac{41}{2}a+8$, $\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{4}{3}a^{5}-\frac{19}{3}a^{4}+2a^{3}+\frac{23}{3}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{7}+\frac{1}{12}a^{6}+\frac{5}{6}a^{5}-3a^{4}-\frac{29}{6}a^{3}+\frac{13}{6}a^{2}+\frac{29}{4}a+\frac{13}{4}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 660.647993664 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{2}\cdot 660.647993664 \cdot 1}{2\sqrt{32788343808}}\approx 1.15228551876$

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

\(\Q(\sqrt{33}) \)

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 ${\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

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 2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.6.6$x^{6} + 3 x + 3$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
\(11\) Copy content Toggle raw display 11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
1.44.2t1.a.a$1$ $ 2^{2} \cdot 11 $ \(\Q(\sqrt{11}) \) $C_2$ (as 2T1) $1$ $1$
* 1.33.2t1.a.a$1$ $ 3 \cdot 11 $ \(\Q(\sqrt{33}) \) $C_2$ (as 2T1) $1$ $1$
2.396.4t3.d.a$2$ $ 2^{2} \cdot 3^{2} \cdot 11 $ 4.0.4752.1 $D_{4}$ (as 4T3) $1$ $-2$
4.42768.6t13.b.a$4$ $ 2^{4} \cdot 3^{5} \cdot 11 $ 6.2.513216.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.1881792.12t34.b.a$4$ $ 2^{6} \cdot 3^{5} \cdot 11^{2}$ 6.2.513216.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5174928.12t34.e.a$4$ $ 2^{4} \cdot 3^{5} \cdot 11^{3}$ 6.2.513216.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.117612.6t13.b.a$4$ $ 2^{2} \cdot 3^{5} \cdot 11^{2}$ 6.2.513216.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
6.2980758528.12t201.a.a$6$ $ 2^{10} \cdot 3^{7} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.3974344704.12t202.a.a$6$ $ 2^{12} \cdot 3^{6} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $2$
* 6.993586176.8t47.a.a$6$ $ 2^{10} \cdot 3^{6} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $2$
6.11923034112.12t200.b.a$6$ $ 2^{12} \cdot 3^{7} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.206...536.16t1294.a.a$9$ $ 2^{18} \cdot 3^{10} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $1$
9.618...608.18t272.a.a$9$ $ 2^{18} \cdot 3^{11} \cdot 11^{3}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $1$
9.274...416.18t273.b.a$9$ $ 2^{18} \cdot 3^{10} \cdot 11^{6}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $1$
9.822...248.18t274.a.a$9$ $ 2^{18} \cdot 3^{11} \cdot 11^{6}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $1$
12.426...632.36t1763.a.a$12$ $ 2^{24} \cdot 3^{15} \cdot 11^{6}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $0$
12.266...352.24t2821.a.a$12$ $ 2^{20} \cdot 3^{15} \cdot 11^{6}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $0$
18.508...784.36t1758.a.a$18$ $ 2^{36} \cdot 3^{22} \cdot 11^{9}$ 8.4.32788343808.3 $S_4\wr C_2$ (as 8T47) $1$ $-2$

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 *.