# Properties

 Label 1.24.2t1.b.a Dimension $1$ Group $C_2$ Conductor $24$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$24$$$$\medspace = 2^{3} \cdot 3$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of $$\Q(\sqrt{-6})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: odd Dirichlet character: $$\displaystyle\left(\frac{-24}{\bullet}\right)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} + 6$$ x^2 + 6 .

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$2 + 2\cdot 5 + 5^{2} + 5^{3} + 2\cdot 5^{4} +O(5^{5})$$ 2 + 2*5 + 5^2 + 5^3 + 2*5^4+O(5^5) $r_{ 2 }$ $=$ $$3 + 2\cdot 5 + 3\cdot 5^{2} + 3\cdot 5^{3} + 2\cdot 5^{4} +O(5^{5})$$ 3 + 2*5 + 3*5^2 + 3*5^3 + 2*5^4+O(5^5)

## Generators of the action on the roots $r_{ 1 }, r_{ 2 }$

 Cycle notation $(1,2)$

## Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.