# Properties

 Label 6.115...287.20t30.a.a Dimension $6$ Group $S_5$ Conductor $1.152\times 10^{13}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_5$ Conductor: $$11517146829287$$$$\medspace = 11^{3} \cdot 2053^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.3.22583.1 Galois orbit size: $1$ Smallest permutation container: 20T30 Parity: odd Determinant: 1.22583.2t1.a.a Projective image: $S_5$ Projective stem field: 5.3.22583.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$16 + 103\cdot 157 + 13\cdot 157^{2} + 151\cdot 157^{3} + 13\cdot 157^{4} +O(157^{5})$$ $r_{ 2 }$ $=$ $$53 + 6\cdot 157 + 17\cdot 157^{2} + 135\cdot 157^{3} + 45\cdot 157^{4} +O(157^{5})$$ $r_{ 3 }$ $=$ $$55 + 69\cdot 157 + 73\cdot 157^{2} + 21\cdot 157^{3} + 114\cdot 157^{4} +O(157^{5})$$ $r_{ 4 }$ $=$ $$64 + 7\cdot 157 + 106\cdot 157^{2} + 21\cdot 157^{3} + 138\cdot 157^{4} +O(157^{5})$$ $r_{ 5 }$ $=$ $$126 + 127\cdot 157 + 103\cdot 157^{2} + 141\cdot 157^{3} + 157^{4} +O(157^{5})$$

## 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 $1$ $1$ $()$ $6$ $10$ $2$ $(1,2)$ $0$ $15$ $2$ $(1,2)(3,4)$ $-2$ $20$ $3$ $(1,2,3)$ $0$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $1$ $20$ $6$ $(1,2,3)(4,5)$ $0$

The blue line marks the conjugacy class containing complex conjugation.