# Properties

 Label 6.0.16807.1 Degree $6$ Signature $[0, 3]$ Discriminant $-16807$ Root discriminant $5.06$ Ramified prime $7$ Class number $1$ Class group trivial Galois group $C_6$ (as 6T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

gp: K = bnfinit(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

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

$$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$

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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $7$ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.a.a$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.6t1.a.a$1$ $7$ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.a.b$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.6t1.a.b$1$ $7$ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$

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 *.