# Properties

 Label 3.61504.12t33.a.b Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 31^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $61504= 2^{6} \cdot 31^{2}$ Artin number field: Splitting field of 5.1.59105344.1 defined by $f= x^{5} - 2 x^{4} + 14 x^{3} - 18 x^{2} + 47 x - 36$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective field: Galois closure of 5.1.59105344.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $54 + 21\cdot 257 + 142\cdot 257^{2} + 138\cdot 257^{3} + 153\cdot 257^{4} +O\left(257^{ 5 }\right)$ $r_{ 2 }$ $=$ $90 + 102\cdot 257 + 212\cdot 257^{2} + 217\cdot 257^{3} + 210\cdot 257^{4} +O\left(257^{ 5 }\right)$ $r_{ 3 }$ $=$ $107 + 6\cdot 257 + 204\cdot 257^{2} + 210\cdot 257^{3} + 181\cdot 257^{4} +O\left(257^{ 5 }\right)$ $r_{ 4 }$ $=$ $117 + 113\cdot 257 + 152\cdot 257^{2} + 253\cdot 257^{3} + 223\cdot 257^{4} +O\left(257^{ 5 }\right)$ $r_{ 5 }$ $=$ $148 + 13\cdot 257 + 60\cdot 257^{2} + 207\cdot 257^{3} +O\left(257^{ 5 }\right)$

### 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 value $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,3,4,5,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.