Properties

Label 8.0.2193125.1
Degree $8$
Signature $[0, 4]$
Discriminant $2193125$
Root discriminant $6.20$
Ramified primes $5, 11, 29$
Class number $1$
Class group trivial
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

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

\(x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1\)  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:  \(2193125\)\(\medspace = 5^{4}\cdot 11^{2}\cdot 29\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $6.20$
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:  $5, 11, 29$
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 \(a\))

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

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:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{7} - 2 a^{6} + 4 a^{5} - 5 a^{4} + 5 a^{3} - 5 a^{2} + 4 a - 2 \),  \( a^{7} - a^{6} + 2 a^{5} - a^{4} + a^{3} - a^{2} + a \),  \( a^{7} - 2 a^{6} + 4 a^{5} - 5 a^{4} + 5 a^{3} - 4 a^{2} + 3 a - 1 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 0.454073872008 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 0.454073872008 \cdot 1}{2\sqrt{2193125}}\approx 0.2389374903087$

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

\(\Q(\sqrt{5}) \), 4.2.275.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.145.2t1.a.a$1$ $ 5 \cdot 29 $ \(\Q(\sqrt{145}) \) $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.29.2t1.a.a$1$ $ 29 $ \(\Q(\sqrt{29}) \) $C_2$ (as 2T1) $1$ $1$
1.1595.2t1.a.a$1$ $ 5 \cdot 11 \cdot 29 $ \(\Q(\sqrt{-1595}) \) $C_2$ (as 2T1) $1$ $-1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
1.55.2t1.a.a$1$ $ 5 \cdot 11 $ \(\Q(\sqrt{-55}) \) $C_2$ (as 2T1) $1$ $-1$
1.319.2t1.a.a$1$ $ 11 \cdot 29 $ \(\Q(\sqrt{-319}) \) $C_2$ (as 2T1) $1$ $-1$
2.1595.4t3.a.a$2$ $ 5 \cdot 11 \cdot 29 $ 4.0.508805.1 $D_{4}$ (as 4T3) $1$ $0$
2.1595.4t3.b.a$2$ $ 5 \cdot 11 \cdot 29 $ 4.0.508805.2 $D_{4}$ (as 4T3) $1$ $0$
2.46255.4t3.a.a$2$ $ 5 \cdot 11 \cdot 29^{2}$ 4.0.508805.4 $D_{4}$ (as 4T3) $1$ $0$
* 2.55.4t3.b.a$2$ $ 5 \cdot 11 $ 4.0.605.1 $D_{4}$ (as 4T3) $1$ $0$
2.145.4t3.a.a$2$ $ 5 \cdot 29 $ 4.4.4205.1 $D_{4}$ (as 4T3) $1$ $2$
2.17545.4t3.a.a$2$ $ 5 \cdot 11^{2} \cdot 29 $ 4.0.508805.3 $D_{4}$ (as 4T3) $1$ $-2$
4.6706975.8t35.f.a$4$ $ 5^{2} \cdot 11 \cdot 29^{3}$ 8.0.2193125.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $-2$
4.12720125.8t29.c.a$4$ $ 5^{3} \cdot 11^{2} \cdot 29^{2}$ 8.0.307827025.1 $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $0$
4.964975.8t35.f.a$4$ $ 5^{2} \cdot 11^{3} \cdot 29 $ 8.0.2193125.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $2$
4.508805.8t29.c.a$4$ $ 5 \cdot 11^{2} \cdot 29^{2}$ 8.0.307827025.1 $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $0$
4.811543975.8t35.f.a$4$ $ 5^{2} \cdot 11^{3} \cdot 29^{3}$ 8.0.2193125.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $2$
* 4.7975.8t35.f.a$4$ $ 5^{2} \cdot 11 \cdot 29 $ 8.0.2193125.1 $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $-2$

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 *.