# Properties

 Label 3.245025.12t33.b Dimension 3 Group $A_5$ Conductor $3^{4} \cdot 5^{2} \cdot 11^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $245025= 3^{4} \cdot 5^{2} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{5} - 3 x^{3} - 4 x^{2} + 6 x + 3$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.245025.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $20 + 103\cdot 131 + 80\cdot 131^{2} + 69\cdot 131^{3} + 64\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 2 }$ $=$ $31 + 94\cdot 131 + 7\cdot 131^{2} + 46\cdot 131^{3} + 44\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 3 }$ $=$ $99 + 106\cdot 131 + 69\cdot 131^{2} + 68\cdot 131^{3} + 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 4 }$ $=$ $116 + 37\cdot 131 + 33\cdot 131^{2} + 24\cdot 131^{3} + 43\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 5 }$ $=$ $127 + 50\cdot 131 + 70\cdot 131^{2} + 53\cdot 131^{3} + 108\cdot 131^{4} +O\left(131^{ 5 }\right)$

### 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.