# Properties

 Label 6.12712961507219.20t30.a.a Dimension 6 Group $S_5$ Conductor $23339^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_5$ Conductor: $12712961507219= 23339^{3}$ Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{2} - 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 20T30 Parity: Odd Determinant: 1.23339.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $14 + 240\cdot 337 + 231\cdot 337^{2} + 234\cdot 337^{3} + 190\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 172\cdot 337 + 42\cdot 337^{2} + 18\cdot 337^{3} + 286\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 3 }$ $=$ $180 + 52\cdot 337 + 192\cdot 337^{2} + 200\cdot 337^{3} + 336\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 4 }$ $=$ $203 + 332\cdot 337 + 260\cdot 337^{2} + 4\cdot 337^{3} + 51\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 5 }$ $=$ $261 + 213\cdot 337 + 283\cdot 337^{2} + 215\cdot 337^{3} + 146\cdot 337^{4} +O\left(337^{ 5 }\right)$

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