# Properties

 Label 5.3e10_5e8.6t15.1 Dimension 5 Group $A_6$ Conductor $3^{10} \cdot 5^{8}$ Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $23066015625= 3^{10} \cdot 5^{8}$ Artin number field: Splitting field of $f= x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ 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 $\Q_{ 127 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $x^{2} + 126 x + 3$
Roots:
 $r_{ 1 }$ $=$ $95 a + 57 + \left(10 a + 13\right)\cdot 127 + \left(10 a + 29\right)\cdot 127^{2} + \left(23 a + 48\right)\cdot 127^{3} + \left(17 a + 2\right)\cdot 127^{4} + \left(2 a + 31\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ $r_{ 2 }$ $=$ $15 + 55\cdot 127 + 61\cdot 127^{2} + 79\cdot 127^{3} + 35\cdot 127^{4} + 7\cdot 127^{5} +O\left(127^{ 6 }\right)$ $r_{ 3 }$ $=$ $32 a + 25 + \left(116 a + 56\right)\cdot 127 + \left(116 a + 28\right)\cdot 127^{2} + \left(103 a + 61\right)\cdot 127^{3} + \left(109 a + 123\right)\cdot 127^{4} + \left(124 a + 15\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ $r_{ 4 }$ $=$ $72 + 79\cdot 127 + 121\cdot 127^{2} + 51\cdot 127^{3} + 67\cdot 127^{4} + 51\cdot 127^{5} +O\left(127^{ 6 }\right)$ $r_{ 5 }$ $=$ $94 a + 59 + \left(40 a + 51\right)\cdot 127 + \left(71 a + 118\right)\cdot 127^{2} + \left(33 a + 88\right)\cdot 127^{3} + \left(63 a + 124\right)\cdot 127^{4} + \left(3 a + 103\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ $r_{ 6 }$ $=$ $33 a + 26 + \left(86 a + 125\right)\cdot 127 + \left(55 a + 21\right)\cdot 127^{2} + \left(93 a + 51\right)\cdot 127^{3} + \left(63 a + 27\right)\cdot 127^{4} + \left(123 a + 44\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$

### 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 $c1$ $1$ $1$ $()$ $5$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $2$ $40$ $3$ $(1,2,3)$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $72$ $5$ $(1,2,3,4,5)$ $0$ $72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.