Properties

Label 8.4.2095196091449344.2
Degree $8$
Signature $[4, 2]$
Discriminant $2^{18}\cdot 13^{4}\cdot 23^{4}$
Root discriminant $82.25$
Ramified primes $2, 13, 23$
Class number $2$
Class group $[2]$
Galois group $(A_4\wr C_2):C_2$ (as 8T45)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 40*x^6 + 16*x^5 + 656*x^4 + 1528*x^3 + 680*x^2 - 1008*x - 2196)
 
gp: K = bnfinit(x^8 - 4*x^7 - 40*x^6 + 16*x^5 + 656*x^4 + 1528*x^3 + 680*x^2 - 1008*x - 2196, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2196, -1008, 680, 1528, 656, 16, -40, -4, 1]);
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196 \)

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:  \(2095196091449344=2^{18}\cdot 13^{4}\cdot 23^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $82.25$
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:  $2, 13, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{8514132} a^{7} + \frac{252859}{4257066} a^{6} + \frac{482137}{4257066} a^{5} - \frac{212147}{2128533} a^{4} + \frac{473257}{4257066} a^{3} - \frac{15134}{2128533} a^{2} + \frac{608090}{2128533} a + \frac{142829}{709511}$

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

Class group and class number

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

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 $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 70518.0197931 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{598}) \)

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 ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.18.57$x^{8} + 12 x^{6} + 36$$8$$1$$18$$A_4\times C_2$$[2, 2, 3]^{3}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$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.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.104.2t1.b.a$1$ $ 2^{3} \cdot 13 $ $x^{2} + 26$ $C_2$ (as 2T1) $1$ $-1$
1.23.2t1.a.a$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
* 1.2392.2t1.a.a$1$ $ 2^{3} \cdot 13 \cdot 23 $ $x^{2} - 598$ $C_2$ (as 2T1) $1$ $1$
2.23.3t2.b.a$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
2.55016.6t3.a.a$2$ $ 2^{3} \cdot 13 \cdot 23^{2}$ $x^{6} + 34 x^{4} - 104 x^{3} + 289 x^{2} - 1768 x - 18824$ $D_{6}$ (as 6T3) $1$ $0$
2.104.3t2.b.a$2$ $ 2^{3} \cdot 13 $ $x^{3} - x - 2$ $S_3$ (as 3T2) $1$ $0$
2.248768.6t3.a.a$2$ $ 2^{6} \cdot 13^{2} \cdot 23 $ $x^{6} + 52 x^{4} + 676 x^{2} - 404248$ $D_{6}$ (as 6T3) $1$ $0$
4.5721664.6t9.a.a$4$ $ 2^{6} \cdot 13^{2} \cdot 23^{2}$ $x^{6} - 2 x^{4} - 50 x^{3} + x^{2} + 50 x + 27$ $S_3^2$ (as 6T9) $1$ $0$
* 6.875918098432.8t45.a.a$6$ $ 2^{15} \cdot 13^{3} \cdot 23^{3}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $2$
6.875918098432.12t161.a.a$6$ $ 2^{15} \cdot 13^{3} \cdot 23^{3}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-2$
9.10657295503622144.18t185.a.a$9$ $ 2^{15} \cdot 13^{3} \cdot 23^{6}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.875918098432.12t165.a.a$9$ $ 2^{15} \cdot 13^{3} \cdot 23^{3}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
9.985288735874613248.18t185.a.a$9$ $ 2^{24} \cdot 13^{6} \cdot 23^{3}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.11988008049386419388416.18t179.a.a$9$ $ 2^{24} \cdot 13^{6} \cdot 23^{6}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
12.767232515160730840858624.24t1503.a.a$12$ $ 2^{30} \cdot 13^{6} \cdot 23^{6}$ $x^{8} - 4 x^{7} - 40 x^{6} + 16 x^{5} + 656 x^{4} + 1528 x^{3} + 680 x^{2} - 1008 x - 2196$ $(A_4\wr C_2):C_2$ (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 *.