# Properties

 Label 5.2e4_13997e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $2^{4} \cdot 13997^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $43875782047568= 2^{4} \cdot 13997^{3}$ Artin number field: Splitting field of $f= x^{5} - 6 x^{3} - 2 x^{2} + 6 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.13997.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $26 + 146\cdot 173 + 116\cdot 173^{2} + 98\cdot 173^{3} + 84\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 2 }$ $=$ $53 + 48\cdot 173 + 14\cdot 173^{2} + 25\cdot 173^{3} + 164\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 3 }$ $=$ $58 + 145\cdot 173 + 138\cdot 173^{2} + 122\cdot 173^{3} + 44\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 4 }$ $=$ $100 + 94\cdot 173 + 35\cdot 173^{2} + 110\cdot 173^{3} + 49\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 5 }$ $=$ $109 + 84\cdot 173 + 40\cdot 173^{2} + 162\cdot 173^{3} + 2\cdot 173^{4} +O\left(173^{ 5 }\right)$

### 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 $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.