# Properties

 Label 5.12588304.6t12.a.a Dimension $5$ Group $A_5$ Conductor $12588304$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $5$ Group: $A_5$ Conductor: $$12588304$$$$\medspace = 2^{4} \cdot 887^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.1.3147076.1 Galois orbit size: $1$ Smallest permutation container: $\PSL(2,5)$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective stem field: 5.1.3147076.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - 2 x^{4} + 9 x^{3} - 12 x^{2} + 30 x + 2$$  .

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$6 + 38\cdot 181 + 181^{2} + 175\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})$$ $r_{ 2 }$ $=$ $$11 + 31\cdot 181 + 125\cdot 181^{2} + 20\cdot 181^{3} + 137\cdot 181^{4} +O(181^{5})$$ $r_{ 3 }$ $=$ $$41 + 166\cdot 181 + 120\cdot 181^{2} + 9\cdot 181^{3} + 10\cdot 181^{4} +O(181^{5})$$ $r_{ 4 }$ $=$ $$143 + 82\cdot 181 + 88\cdot 181^{2} + 177\cdot 181^{3} + 103\cdot 181^{4} +O(181^{5})$$ $r_{ 5 }$ $=$ $$163 + 43\cdot 181 + 26\cdot 181^{2} + 160\cdot 181^{3} + 166\cdot 181^{4} +O(181^{5})$$

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

The blue line marks the conjugacy class containing complex conjugation.