# Properties

 Label 5.8342823647672513.6t14.a.a Dimension 5 Group $S_5$ Conductor $202817^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $8342823647672513= 202817^{3}$ Artin number field: Splitting field of 5.5.202817.1 defined by $f= x^{5} - 2 x^{4} - 4 x^{3} + 5 x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.202817.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.5.202817.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $6 + 27\cdot 347 + 187\cdot 347^{2} + 57\cdot 347^{3} + 183\cdot 347^{4} +O\left(347^{ 5 }\right)$ $r_{ 2 }$ $=$ $131 + 215\cdot 347 + 18\cdot 347^{2} + 339\cdot 347^{3} + 234\cdot 347^{4} +O\left(347^{ 5 }\right)$ $r_{ 3 }$ $=$ $178 + 36\cdot 347 + 130\cdot 347^{2} + 188\cdot 347^{3} + 279\cdot 347^{4} +O\left(347^{ 5 }\right)$ $r_{ 4 }$ $=$ $190 + 168\cdot 347 + 193\cdot 347^{2} + 36\cdot 347^{3} + 18\cdot 347^{4} +O\left(347^{ 5 }\right)$ $r_{ 5 }$ $=$ $191 + 246\cdot 347 + 164\cdot 347^{2} + 72\cdot 347^{3} + 325\cdot 347^{4} +O\left(347^{ 5 }\right)$

### 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.