# Properties

 Label 2.324.3t2.a Dimension $2$ Group $S_3$ Conductor $324$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$324$$$$\medspace = 2^{2} \cdot 3^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.419904.2 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 6.0.419904.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{2} + 7 x + 2$
Roots:
 $r_{ 1 }$ $=$ $2 a + 2 + \left(5 a + 10\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + 9\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 a + 10 + \left(5 a + 6\right)\cdot 11 + \left(2 a + 1\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $3 a + 1 + \left(6 a + 5\right)\cdot 11 + \left(6 a + 7\right)\cdot 11^{2} + \left(4 a + 7\right)\cdot 11^{3} + \left(3 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a + 8 + \left(10 a + 6\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(3 a + 9\right)\cdot 11^{3} + \left(7 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 a + 10 + 9\cdot 11 + \left(4 a + 3\right)\cdot 11^{2} + \left(7 a + 6\right)\cdot 11^{3} + \left(3 a + 10\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 6 }$ $=$ $8 a + 2 + \left(4 a + 5\right)\cdot 11 + \left(4 a + 5\right)\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + \left(7 a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,3,4)(2,5,6)$ $(1,2)(3,6)(4,5)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)(3,6)(4,5)$ $0$ $2$ $3$ $(1,3,4)(2,5,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.