# Properties

 Label 4.24048.6t13.a Dimension $4$ Group $C_3^2:D_4$ Conductor $24048$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$24048$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 167$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.4.288576.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: odd Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.4.288576.1

## 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:
 $r_{ 1 }$ $=$ $7 a + 1 + 10 a\cdot 11 + \left(10 a + 4\right)\cdot 11^{2} + \left(a + 8\right)\cdot 11^{3} + \left(7 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 + 11 + 6\cdot 11^{2} + 10\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a + 4 + \left(6 a + 9\right)\cdot 11 + \left(8 a + 1\right)\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $9 + 10\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 5 }$ $=$ $4 a + 7 + 2\cdot 11 + 4\cdot 11^{2} + \left(9 a + 5\right)\cdot 11^{3} + \left(3 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 6 }$ $=$ $7 a + 9 + \left(4 a + 8\right)\cdot 11 + \left(2 a + 7\right)\cdot 11^{2} + \left(4 a + 4\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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