# Properties

 Label 3.3e6_11e2.12t33.1 Dimension 3 Group $A_5$ Conductor $3^{6} \cdot 11^{2}$ Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $88209= 3^{6} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16 x + 3$
Roots:
 $r_{ 1 }$ $=$ $15 a + 8 + \left(3 a + 16\right)\cdot 17 + \left(13 a + 15\right)\cdot 17^{2} + \left(12 a + 10\right)\cdot 17^{3} + \left(4 a + 3\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 13\cdot 17 + 14\cdot 17^{2} + 16\cdot 17^{3} + 7\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $2 a + 6 + \left(13 a + 5\right)\cdot 17 + \left(3 a + 8\right)\cdot 17^{2} + \left(4 a + 10\right)\cdot 17^{3} + \left(12 a + 12\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 a + 9 + \left(12 a + 16\right)\cdot 17 + \left(11 a + 14\right)\cdot 17^{2} + \left(4 a + 9\right)\cdot 17^{3} + \left(a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 5 }$ $=$ $4 a + 5 + \left(4 a + 16\right)\cdot 17 + \left(5 a + 13\right)\cdot 17^{2} + \left(12 a + 2\right)\cdot 17^{3} + \left(15 a + 3\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.