# Properties

 Label 5.7e4_283e2.6t15.1 Dimension 5 Group $A_6$ Conductor $7^{4} \cdot 283^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $192293689= 7^{4} \cdot 283^{2}$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ 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 $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $x^{2} + 49 x + 2$
Roots:
 $r_{ 1 }$ $=$ $32 + 52\cdot 53 + 38\cdot 53^{2} + 19\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $40 a + 5 + \left(5 a + 34\right)\cdot 53 + \left(3 a + 52\right)\cdot 53^{2} + \left(49 a + 16\right)\cdot 53^{3} + \left(4 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 + 37\cdot 53 + 36\cdot 53^{2} + 24\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 a + 6 + \left(47 a + 17\right)\cdot 53 + \left(49 a + 6\right)\cdot 53^{2} + \left(3 a + 51\right)\cdot 53^{3} + \left(48 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 5 }$ $=$ $22 a + 35 + \left(34 a + 30\right)\cdot 53 + \left(28 a + 51\right)\cdot 53^{2} + \left(36 a + 43\right)\cdot 53^{3} + \left(50 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 6 }$ $=$ $31 a + 17 + \left(18 a + 40\right)\cdot 53 + \left(24 a + 25\right)\cdot 53^{2} + \left(16 a + 2\right)\cdot 53^{3} + \left(2 a + 5\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

### 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.