# Properties

 Label 2.648.3t2.a.a Dimension 2 Group $S_3$ Conductor $2^{3} \cdot 3^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $648= 2^{3} \cdot 3^{4}$ Artin number field: Splitting field of 6.0.3359232.4 defined by $f= x^{6} - 6 x^{4} + 9 x^{2} + 8$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Odd Determinant: 1.8.2t1.b.a Projective image: $S_3$ Projective field: Galois closure of 6.0.3359232.4

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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