# Properties

 Label 4.13401.6t13.b Dimension $4$ Group $C_3^2:D_4$ Conductor $13401$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$13401$$$$\medspace = 3^{2} \cdot 1489$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.40203.1 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 6.0.40203.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $8 a + 4 + \left(3 a + 8\right)\cdot 19 + 17\cdot 19^{2} + \left(13 a + 13\right)\cdot 19^{3} + \left(11 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $15 + 10\cdot 19 + 17\cdot 19^{2} + 15\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 + 13\cdot 19 + 14\cdot 19^{2} + 9\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 12 + \left(15 a + 3\right)\cdot 19 + \left(18 a + 14\right)\cdot 19^{2} + \left(5 a + 7\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 a + 14 + \left(9 a + 18\right)\cdot 19 + \left(7 a + 6\right)\cdot 19^{2} + 16 a\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $12 a + 2 + \left(9 a + 2\right)\cdot 19 + \left(11 a + 5\right)\cdot 19^{2} + \left(2 a + 9\right)\cdot 19^{3} + \left(16 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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