# Properties

 Label 4.6208.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $6208$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$6208$$$$\medspace = 2^{6} \cdot 97$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.49664.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Determinant: 1.97.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.49664.1

## Defining polynomial

 $f(x)$ $=$ $x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} - 1$.

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$

Roots:
 $r_{ 1 }$ $=$ $12 + 44\cdot 47 + 40\cdot 47^{2} + 2\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 35 + \left(41 a + 3\right)\cdot 47 + \left(21 a + 27\right)\cdot 47^{2} + \left(5 a + 46\right)\cdot 47^{3} + \left(27 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 + 19\cdot 47 + 13\cdot 47^{2} + 11\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $44 a + 41 + \left(5 a + 35\right)\cdot 47 + \left(25 a + 29\right)\cdot 47^{2} + \left(41 a + 35\right)\cdot 47^{3} + \left(19 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $17 a + 29 + \left(18 a + 32\right)\cdot 47 + \left(3 a + 20\right)\cdot 47^{2} + \left(34 a + 13\right)\cdot 47^{3} + \left(13 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $30 a + 16 + \left(28 a + 5\right)\cdot 47 + \left(43 a + 9\right)\cdot 47^{2} + \left(12 a + 31\right)\cdot 47^{3} + \left(33 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

## 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 value $1$ $1$ $()$ $4$ $6$ $2$ $(1,3)(2,5)(4,6)$ $0$ $6$ $2$ $(2,4)$ $2$ $9$ $2$ $(2,4)(5,6)$ $0$ $4$ $3$ $(1,2,4)$ $1$ $4$ $3$ $(1,2,4)(3,5,6)$ $-2$ $18$ $4$ $(1,3)(2,6,4,5)$ $0$ $12$ $6$ $(1,5,2,6,4,3)$ $0$ $12$ $6$ $(2,4)(3,5,6)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.