# Properties

 Label 2.524.3t2.a.a Dimension $2$ Group $S_3$ Conductor $524$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$524$$$$\medspace = 2^{2} \cdot 131$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 3.1.524.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.131.2t1.a.a Projective image: $S_3$ Projective stem field: 3.1.524.1

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - x^{2} + 3 x - 5$$  .

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

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

## Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$ $(1,2)$

## Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.