# Properties

 Label 2.0.127.1 Degree $2$ Signature $[0, 1]$ Discriminant $-127$ Root discriminant $11.27$ Ramified prime $127$ Class number $5$ Class group $[5]$ Galois group $C_2$ (as 2T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 32)

gp: K = bnfinit(x^2 - x + 32, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -1, 1]);

$$x^{2} - x + 32$$

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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.127.2t1.a.a$1$ $127$ $x^{2} - x + 32$ $C_2$ (as 2T1) $1$ $-1$

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 *.