# Properties

 Label 5.2e4_3e6_13e2.6t15.2 Dimension 5 Group $A_6$ Conductor $2^{4} \cdot 3^{6} \cdot 13^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $1971216= 2^{4} \cdot 3^{6} \cdot 13^{2}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 2$ 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 $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $x^{2} + 70 x + 5$
Roots:
 $r_{ 1 }$ $=$ $61 a + 28 + \left(14 a + 25\right)\cdot 73 + \left(57 a + 30\right)\cdot 73^{2} + \left(34 a + 65\right)\cdot 73^{3} + \left(15 a + 71\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $60 + 46\cdot 73 + 62\cdot 73^{2} + 17\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 a + 65 + \left(58 a + 8\right)\cdot 73 + \left(15 a + 41\right)\cdot 73^{2} + \left(38 a + 39\right)\cdot 73^{3} + \left(57 a + 10\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $61 + 21\cdot 73 + 68\cdot 73^{2} + 67\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $72 a + 42 + \left(41 a + 67\right)\cdot 73 + \left(27 a + 60\right)\cdot 73^{2} + \left(44 a + 70\right)\cdot 73^{3} + \left(63 a + 20\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 6 }$ $=$ $a + 39 + \left(31 a + 48\right)\cdot 73 + \left(45 a + 28\right)\cdot 73^{2} + \left(28 a + 30\right)\cdot 73^{3} + \left(9 a + 21\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$

### 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.