# Properties

 Label 5.2e6_44171e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $2^{6} \cdot 44171^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $5515586099981504= 2^{6} \cdot 44171^{3}$ Artin number field: Splitting field of $f= x^{5} - 6 x^{3} - x^{2} + 5 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.2e2_44171.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 23 + 133\cdot 269 + 44\cdot 269^{2} + 33\cdot 269^{3} + 60\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 2 } &= 136 + 243\cdot 269 + 187\cdot 269^{2} + 260\cdot 269^{3} + 125\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 3 } &= 176 + 64\cdot 269 + 77\cdot 269^{2} + 41\cdot 269^{3} + 77\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 4 } &= 224 + 37\cdot 269 + 111\cdot 269^{2} + 8\cdot 269^{3} + 49\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 5 } &= 248 + 58\cdot 269 + 117\cdot 269^{2} + 194\cdot 269^{3} + 225\cdot 269^{4} +O\left(269^{ 5 }\right) \\ \end{aligned}

### 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.