# Properties

 Label 4.23488.6t13.a.a Dimension $4$ Group $C_3^2:D_4$ Conductor $23488$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$23488$$$$\medspace = 2^{6} \cdot 367$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.4.187904.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: odd Determinant: 1.367.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.4.187904.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$

Roots:
 $r_{ 1 }$ $=$ $8 a + 13 + \left(6 a + 16\right)\cdot 23 + \left(4 a + 20\right)\cdot 23^{2} + \left(13 a + 14\right)\cdot 23^{3} + \left(17 a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 7 + \left(22 a + 15\right)\cdot 23 + \left(2 a + 2\right)\cdot 23^{2} + \left(6 a + 9\right)\cdot 23^{3} + \left(22 a + 20\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 a + 13 + 10\cdot 23 + \left(20 a + 9\right)\cdot 23^{2} + \left(16 a + 18\right)\cdot 23^{3} + 12\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $9 + 23 + 2\cdot 23^{2} + 3\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $21 + 3\cdot 23 + 11\cdot 23^{2} + 9\cdot 23^{3} + 9\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $15 a + 6 + \left(16 a + 21\right)\cdot 23 + \left(18 a + 22\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.