Properties

Label 8.0.562...416.1
Degree $8$
Signature $[0, 4]$
Discriminant $5.620\times 10^{51}$
Root discriminant \(2\,942\,505.85\)
Ramified primes see page
Class number not computed
Class group not computed
Galois group $A_8$ (as 8T49)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210825316)
 
gp: K = bnfinit(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210825316, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![210825316, -6144, -8960, -7168, -3360, -896, -112, 0, 1]);
 

\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825316 \) 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:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(5620030628480917722529796097626934820084447048892416\) \(\medspace = 2^{18}\cdot 23^{6}\cdot 43^{6}\cdot 137^{6}\cdot 389^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(2\,942\,505.85\)
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:   \(2\), \(23\), \(43\), \(137\), \(389\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{40}a^{7}+\frac{1}{20}a^{6}+\frac{1}{20}a^{5}-\frac{1}{20}a^{4}-\frac{7}{20}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{3}{10}$ 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

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$A_8$ (as 8T49):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 15 siblings: Deg 15, Deg 15
Degree 28 sibling: Deg 28
Degree 35 sibling: Deg 35

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ R ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.8.18.6$x^{8} + 16 x^{4} + 4$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.7.6.1$x^{7} - 23$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
\(137\) Copy content Toggle raw display $\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.7.6.1$x^{7} - 137$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(389\) Copy content Toggle raw display $\Q_{389}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $7$$7$$1$$6$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 7.562...416.8t49.a.a$7$ $ 2^{18} \cdot 23^{6} \cdot 43^{6} \cdot 137^{6} \cdot 389^{6}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $-1$
14.123...176.15t72.a.a$14$ $ 2^{28} \cdot 23^{12} \cdot 43^{12} \cdot 137^{12} \cdot 389^{12}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $6$
20.693...216.28t433.a.a$20$ $ 2^{46} \cdot 23^{18} \cdot 43^{18} \cdot 137^{18} \cdot 389^{18}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $4$
21.113...944.56.a.a$21$ $ 2^{60} \cdot 23^{18} \cdot 43^{18} \cdot 137^{18} \cdot 389^{18}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $-3$
21.113...944.336.a.a$21$ $ 2^{60} \cdot 23^{18} \cdot 43^{18} \cdot 137^{18} \cdot 389^{18}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $0$ $-3$
21.113...944.336.a.b$21$ $ 2^{60} \cdot 23^{18} \cdot 43^{18} \cdot 137^{18} \cdot 389^{18}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $0$ $-3$
28.638...704.56.a.a$28$ $ 2^{78} \cdot 23^{24} \cdot 43^{24} \cdot 137^{24} \cdot 389^{24}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $-4$
35.143...456.70.a.a$35$ $ 2^{98} \cdot 23^{30} \cdot 43^{30} \cdot 137^{30} \cdot 389^{30}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $3$
45.193...312.336.a.a$45$ $ 2^{130} \cdot 23^{39} \cdot 43^{39} \cdot 137^{39} \cdot 389^{39}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $0$ $-3$
45.193...312.336.a.b$45$ $ 2^{130} \cdot 23^{39} \cdot 43^{39} \cdot 137^{39} \cdot 389^{39}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $0$ $-3$
56.995...496.105.a.a$56$ $ 2^{144} \cdot 23^{48} \cdot 43^{48} \cdot 137^{48} \cdot 389^{48}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $8$
64.916...024.168.a.a$64$ $ 2^{176} \cdot 23^{54} \cdot 43^{54} \cdot 137^{54} \cdot 389^{54}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $0$
70.514...984.120.a.a$70$ $ 2^{194} \cdot 23^{60} \cdot 43^{60} \cdot 137^{60} \cdot 389^{60}$ 8.0.5620030628480917722529796097626934820084447048892416.1 $A_8$ (as 8T49) $1$ $-2$

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 *.