# Properties

 Label 3.40000.12t33.e Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 5^{4}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $40000= 2^{6} \cdot 5^{4}$ Artin number field: Splitting field of $f= x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.25000000.4

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $1 + 35\cdot 197 + 165\cdot 197^{2} + 58\cdot 197^{3} + 14\cdot 197^{4} +O\left(197^{ 5 }\right)$ $r_{ 2 }$ $=$ $85 + 40\cdot 197 + 153\cdot 197^{2} + 25\cdot 197^{3} + 182\cdot 197^{4} +O\left(197^{ 5 }\right)$ $r_{ 3 }$ $=$ $154 + 93\cdot 197 + 186\cdot 197^{2} + 155\cdot 197^{3} + 60\cdot 197^{4} +O\left(197^{ 5 }\right)$ $r_{ 4 }$ $=$ $164 + 150\cdot 197 + 104\cdot 197^{2} + 75\cdot 197^{3} + 110\cdot 197^{4} +O\left(197^{ 5 }\right)$ $r_{ 5 }$ $=$ $187 + 73\cdot 197 + 178\cdot 197^{2} + 77\cdot 197^{3} + 26\cdot 197^{4} +O\left(197^{ 5 }\right)$

### 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.