# Properties

 Label 8.0.16777216.1 Degree $8$ Signature $[0, 4]$ Discriminant $16777216$ Root discriminant $8.00$ Ramified prime $2$ Class number $1$ Class group trivial Galois group $C_4\times C_2$ (as 8T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 1)

gp: K = bnfinit(x^8 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 1]);

$$x^{8} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.8.2t1.b.a$1$ $2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.16.4t1.a.a$1$ $2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
* 1.16.4t1.b.a$1$ $2^{4}$ $x^{4} + 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $-1$
* 1.8.2t1.a.a$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.16.4t1.a.b$1$ $2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
* 1.16.4t1.b.b$1$ $2^{4}$ $x^{4} + 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $-1$

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 *.