Properties

 Label 4.0.2197.1 Degree $4$ Signature $[0, 2]$ Discriminant $2197$ Root discriminant $6.85$ Ramified prime $13$ Class number $1$ Class group trivial Galois group $C_4$ (as 4T1)

Related objects

Show commands for: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 4, 2, -1, 1]);

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

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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.13.4t1.a.a$1$ $13$ $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ $C_4$ (as 4T1) $0$ $-1$
* 1.13.2t1.a.a$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.13.4t1.a.b$1$ $13$ $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ $C_4$ (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 *.