Properties

Label 8.0.1492101.1
Degree $8$
Signature $[0, 4]$
Discriminant $1492101$
Root discriminant $5.91$
Ramified primes $3, 13, 109$
Class number $1$
Class group trivial
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^6 - 3*x^5 + x^4 - 2*x^3 + 3*x^2 + 1)
 
gp: K = bnfinit(x^8 - x^7 + x^6 - 3*x^5 + x^4 - 2*x^3 + 3*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 3, -2, 1, -3, 1, -1, 1]);
 

\( x^{8} - x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + 3 x^{2} + 1 \)

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:  \(1492101\)\(\medspace = 3^{4}\cdot 13^{2}\cdot 109\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $5.91$
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:  $3, 13, 109$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

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

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:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a^{6} + a^{4} - a^{3} - a^{2} - 2 a \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{5} + a^{3} - 2 a^{2} - a - 2 \),  \( a^{7} + a^{5} - a^{4} - a^{3} - 2 a^{2} \),  \( a^{7} - a^{6} + 2 a^{5} - 3 a^{4} + 2 a^{3} - 3 a^{2} + 2 a - 1 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1.04325557006 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.04325557006 \cdot 1}{6\sqrt{1492101}}\approx 0.221850199947$

Galois group

$D_4^2.C_2$ (as 8T35):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$
Character table for $C_2 \wr C_2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.39.2t1.a.a$1$ $ 3 \cdot 13 $ \(\Q(\sqrt{-39}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.327.2t1.a.a$1$ $ 3 \cdot 109 $ \(\Q(\sqrt{-327}) \) $C_2$ (as 2T1) $1$ $-1$
1.4251.2t1.a.a$1$ $ 3 \cdot 13 \cdot 109 $ \(\Q(\sqrt{-4251}) \) $C_2$ (as 2T1) $1$ $-1$
1.1417.2t1.a.a$1$ $ 13 \cdot 109 $ \(\Q(\sqrt{1417}) \) $C_2$ (as 2T1) $1$ $1$
1.109.2t1.a.a$1$ $ 109 $ \(\Q(\sqrt{109}) \) $C_2$ (as 2T1) $1$ $1$
2.55263.4t3.b.a$2$ $ 3 \cdot 13^{2} \cdot 109 $ 4.0.165789.1 $D_{4}$ (as 4T3) $1$ $0$
2.463359.4t3.a.a$2$ $ 3 \cdot 13 \cdot 109^{2}$ 4.2.6023667.3 $D_{4}$ (as 4T3) $1$ $0$
2.4251.4t3.d.a$2$ $ 3 \cdot 13 \cdot 109 $ 4.0.12753.2 $D_{4}$ (as 4T3) $1$ $0$
2.4251.4t3.c.a$2$ $ 3 \cdot 13 \cdot 109 $ 4.0.12753.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.39.4t3.b.a$2$ $ 3 \cdot 13 $ 4.2.507.1 $D_{4}$ (as 4T3) $1$ $0$
2.327.4t3.c.a$2$ $ 3 \cdot 109 $ 4.0.981.1 $D_{4}$ (as 4T3) $1$ $0$
4.54213003.8t29.a.a$4$ $ 3^{3} \cdot 13^{2} \cdot 109^{2}$ 8.0.27485992521.4 $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$
4.2155257.8t35.e.a$4$ $ 3^{2} \cdot 13^{3} \cdot 109 $ 8.0.1492101.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.151518393.8t35.e.a$4$ $ 3^{2} \cdot 13 \cdot 109^{3}$ 8.0.1492101.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
* 4.12753.8t35.e.a$4$ $ 3^{2} \cdot 13 \cdot 109 $ 8.0.1492101.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.25606608417.8t35.e.a$4$ $ 3^{2} \cdot 13^{3} \cdot 109^{3}$ 8.0.1492101.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.6023667.8t29.a.a$4$ $ 3 \cdot 13^{2} \cdot 109^{2}$ 8.0.27485992521.4 $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$

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 *.