# Properties

 Label 8.2e18_3e16.36t555.1 Dimension 8 Group $A_6$ Conductor $2^{18} \cdot 3^{16}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $8$ Group: $A_6$ Conductor: $11284439629824= 2^{18} \cdot 3^{16}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_6$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $x^{2} + 63 x + 2$
Roots:
 $r_{ 1 }$ $=$ $65 a + 49 + \left(29 a + 24\right)\cdot 67 + \left(44 a + 10\right)\cdot 67^{2} + \left(25 a + 60\right)\cdot 67^{3} + \left(31 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 a + 41 + \left(37 a + 12\right)\cdot 67 + \left(22 a + 24\right)\cdot 67^{2} + \left(41 a + 51\right)\cdot 67^{3} + \left(35 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 + 58\cdot 67 + 31\cdot 67^{2} + 61\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $47 + 66\cdot 67 + 45\cdot 67^{2} + 42\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 a + 45 + \left(18 a + 52\right)\cdot 67 + \left(36 a + 47\right)\cdot 67^{2} + \left(16 a + 44\right)\cdot 67^{3} + \left(58 a + 1\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 6 }$ $=$ $62 a + 65 + \left(48 a + 52\right)\cdot 67 + \left(30 a + 40\right)\cdot 67^{2} + \left(50 a + 7\right)\cdot 67^{3} + \left(8 a + 17\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

### 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 $c1$ $c2$ $1$ $1$ $()$ $8$ $8$ $45$ $2$ $(1,2)(3,4)$ $0$ $0$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $-1$ $40$ $3$ $(1,2,3)$ $-1$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $0$ $72$ $5$ $(1,2,3,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $72$ $5$ $(1,3,4,5,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.