# Properties

 Label 4.4338889.5t4.a.a Dimension $4$ Group $A_5$ Conductor $4338889$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $A_5$ Conductor: $$4338889$$$$\medspace = 2083^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.1.4338889.1 Galois orbit size: $1$ Smallest permutation container: $A_5$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective stem field: 5.1.4338889.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} + 5 x^{3} + 11 x^{2} + 4 x - 1$$  .

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$47 + 194\cdot 241 + 5\cdot 241^{2} + 200\cdot 241^{3} + 188\cdot 241^{4} +O(241^{5})$$ $r_{ 2 }$ $=$ $$88 + 84\cdot 241 + 45\cdot 241^{2} + 31\cdot 241^{3} + 45\cdot 241^{4} +O(241^{5})$$ $r_{ 3 }$ $=$ $$175 + 27\cdot 241 + 12\cdot 241^{2} + 65\cdot 241^{3} + 218\cdot 241^{4} +O(241^{5})$$ $r_{ 4 }$ $=$ $$188 + 88\cdot 241 + 233\cdot 241^{2} + 186\cdot 241^{3} + 5\cdot 241^{4} +O(241^{5})$$ $r_{ 5 }$ $=$ $$226 + 86\cdot 241 + 185\cdot 241^{2} + 239\cdot 241^{3} + 23\cdot 241^{4} +O(241^{5})$$

## 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 $1$ $1$ $()$ $4$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $12$ $5$ $(1,2,3,4,5)$ $-1$ $12$ $5$ $(1,3,4,5,2)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.