# Properties

 Label 5.2e13_5e9_7e5.6t14.2c1 Dimension 5 Group $S_5$ Conductor $2^{13} \cdot 5^{9} \cdot 7^{5}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $268912000000000= 2^{13} \cdot 5^{9} \cdot 7^{5}$ Artin number field: Splitting field of $f=x^{5} - 100 x^{3} - 500 x^{2} - 75 x + 55080$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 6T14 Parity: Even Determinant: 1.2e3_5_7.2t1.2c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= -37113 +O\left(11^{ 5 }\right) \\ r_{ 2 } &= -22889 +O\left(11^{ 5 }\right) \\ r_{ 3 } &= -52917 +O\left(11^{ 5 }\right) \\ r_{ 4 } &= -52155 +O\left(11^{ 5 }\right) \\ r_{ 5 } &= 4023 +O\left(11^{ 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.