# Properties

 Label 4.13941.6t13.b Dimension $4$ Group $C_3^2:D_4$ Conductor $13941$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$13941$$$$\medspace = 3^{2} \cdot 1549$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.41823.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.41823.1

## 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:
 $r_{ 1 }$ $=$ $3 + 6\cdot 7 + 3\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 + 6\cdot 7 + 3\cdot 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 3 }$ $=$ $6 a + \left(2 a + 2\right)\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(4 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a + 3 + \left(6 a + 3\right)\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 5 }$ $=$ $a + 6 + \left(4 a + 5\right)\cdot 7 + \left(4 a + 4\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(2 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 6 }$ $=$ $a + 2 + 4\cdot 7 + 5 a\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)$ $1$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.