# Properties

 Label 5.3e8_7e2_23e2.6t15.1 Dimension 5 Group $A_6$ Conductor $3^{8} \cdot 7^{2} \cdot 23^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $170067681= 3^{8} \cdot 7^{2} \cdot 23^{2}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_6$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $27 + 19\cdot 31 + 20\cdot 31^{2} + 2\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 a + 29 + \left(23 a + 28\right)\cdot 31 + 11 a\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 5\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(13 a + 15\right)\cdot 31^{3} + \left(29 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $19 a + 6 + \left(16 a + 15\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + \left(17 a + 25\right)\cdot 31^{3} + \left(a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $11 a + 7 + \left(7 a + 25\right)\cdot 31 + 19 a\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(30 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $5$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $40$ $3$ $(1,2,3)$ $2$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $72$ $5$ $(1,2,3,4,5)$ $0$ $72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.