# Properties

 Label 4.5164.5t5.a.a Dimension $4$ Group $S_5$ Conductor $5164$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $$5164$$$$\medspace = 2^{2} \cdot 1291$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.5164.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.5164.2t1.a.a Projective image: $S_5$ Projective stem field: Galois closure of 5.1.5164.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$57 + 67\cdot 283 + 157\cdot 283^{2} + 9\cdot 283^{3} + 61\cdot 283^{4} +O(283^{5})$$ 57 + 67*283 + 157*283^2 + 9*283^3 + 61*283^4+O(283^5) $r_{ 2 }$ $=$ $$154 + 11\cdot 283 + 124\cdot 283^{2} + 196\cdot 283^{3} + 198\cdot 283^{4} +O(283^{5})$$ 154 + 11*283 + 124*283^2 + 196*283^3 + 198*283^4+O(283^5) $r_{ 3 }$ $=$ $$180 + 118\cdot 283 + 256\cdot 283^{2} + 131\cdot 283^{3} + 99\cdot 283^{4} +O(283^{5})$$ 180 + 118*283 + 256*283^2 + 131*283^3 + 99*283^4+O(283^5) $r_{ 4 }$ $=$ $$198 + 254\cdot 283 + 20\cdot 283^{2} + 86\cdot 283^{3} + 137\cdot 283^{4} +O(283^{5})$$ 198 + 254*283 + 20*283^2 + 86*283^3 + 137*283^4+O(283^5) $r_{ 5 }$ $=$ $$261 + 113\cdot 283 + 7\cdot 283^{2} + 142\cdot 283^{3} + 69\cdot 283^{4} +O(283^{5})$$ 261 + 113*283 + 7*283^2 + 142*283^3 + 69*283^4+O(283^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.