# Properties

 Label 4.0.125.1 Degree $4$ Signature $[0, 2]$ Discriminant $125$ Root discriminant $3.34$ Ramified prime $5$ Class number $1$ Class group trivial Galois group $C_4$ (as 4T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

This is the quartic field with Galois group $C_4$ with the smallest absolute discriminant.

## Normalizeddefining polynomial

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

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

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

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

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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.a.a$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.a.b$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (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 *.