# Properties

 Label 3.2e6_41e2.12t33.1c2 Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 41^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $107584= 2^{6} \cdot 41^{2}$ Artin number field: Splitting field of $f= x^{5} - x^{4} - 16 x^{3} + 36 x^{2} + 267 x - 319$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $3 + 20\cdot 67 + 55\cdot 67^{2} + 61\cdot 67^{3} + 46\cdot 67^{4} + 56\cdot 67^{5} +O\left(67^{ 6 }\right)$ $r_{ 2 }$ $=$ $28 + 53\cdot 67 + 47\cdot 67^{2} + 14\cdot 67^{3} + 9\cdot 67^{4} + 67^{5} +O\left(67^{ 6 }\right)$ $r_{ 3 }$ $=$ $49 + 62\cdot 67 + 53\cdot 67^{2} + 7\cdot 67^{3} + 11\cdot 67^{4} + 55\cdot 67^{5} +O\left(67^{ 6 }\right)$ $r_{ 4 }$ $=$ $56 + 21\cdot 67 + 25\cdot 67^{2} + 32\cdot 67^{3} + 11\cdot 67^{4} + 22\cdot 67^{5} +O\left(67^{ 6 }\right)$ $r_{ 5 }$ $=$ $66 + 42\cdot 67 + 18\cdot 67^{2} + 17\cdot 67^{3} + 55\cdot 67^{4} + 65\cdot 67^{5} +O\left(67^{ 6 }\right)$

### 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.