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Properties

Label 8.0.1287913472.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{18}\cdot 17^{3}$
Root discriminant $13.76$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $Z_8 : Z_8^\times$ (as 8T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68, 0, -4, 0, 10, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 + 10*x^4 - 4*x^2 + 68)
 
gp: K = bnfinit(x^8 - 2*x^6 + 10*x^4 - 4*x^2 + 68, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{6} + 10 x^{4} - 4 x^{2} + 68 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1287913472=2^{18}\cdot 17^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.76$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{206} a^{6} + \frac{19}{206} a^{4} + \frac{50}{103} a^{2} + \frac{18}{103}$, $\frac{1}{206} a^{7} + \frac{19}{206} a^{5} + \frac{50}{103} a^{3} + \frac{18}{103} a$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{5}{206} a^{6} + \frac{4}{103} a^{4} - \frac{44}{103} a^{2} + \frac{13}{103} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{3}{103} a^{6} + \frac{11}{206} a^{4} - \frac{9}{103} a^{2} + \frac{5}{103} \),  \( \frac{9}{103} a^{7} + \frac{3}{103} a^{6} - \frac{35}{103} a^{5} - \frac{46}{103} a^{4} + \frac{76}{103} a^{3} + \frac{94}{103} a^{2} - \frac{88}{103} a - \frac{407}{103} \),  \( \frac{19}{206} a^{7} + \frac{3}{103} a^{6} - \frac{51}{206} a^{5} + \frac{57}{103} a^{4} + \frac{126}{103} a^{3} - \frac{112}{103} a^{2} + \frac{33}{103} a + \frac{417}{103} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47.1972100966 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 8T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $Z_8 : Z_8^\times$
Character table for $Z_8 : Z_8^\times$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1

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

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed
Arithmetically equvalently sibling: 8.0.1287913472.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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

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];
 
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]])
 

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_17.2t1.1c1$1$ $ 2^{3} \cdot 17 $ $x^{2} - 34$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_17.2t1.2c1$1$ $ 2^{3} \cdot 17 $ $x^{2} + 34$ $C_2$ (as 2T1) $1$ $-1$
1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e2_17.2t1.1c1$1$ $ 2^{2} \cdot 17 $ $x^{2} + 17$ $C_2$ (as 2T1) $1$ $-1$
1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_17.4t3.1c1$2$ $ 2^{2} \cdot 17 $ $x^{4} + x^{2} - 2 x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.2e6_17.4t3.2c1$2$ $ 2^{6} \cdot 17 $ $x^{4} - 8 x^{2} + 17$ $D_{4}$ (as 4T3) $1$ $0$
* 4.2e14_17e2.8t15.2c1$4$ $ 2^{14} \cdot 17^{2}$ $x^{8} - 2 x^{6} + 10 x^{4} - 4 x^{2} + 68$ $Z_8 : Z_8^\times$ (as 8T15) $1$ $0$

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 *.