# Properties

 Label 5.7e3_11e3_1621e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $7^{3} \cdot 11^{3} \cdot 1621^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $1944559427246513= 7^{3} \cdot 11^{3} \cdot 1621^{3}$ Artin number field: Splitting field of $f= x^{5} - 7 x^{3} - 6 x^{2} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.7_11_1621.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 44 + 244\cdot 457 + 322\cdot 457^{2} + 104\cdot 457^{3} + 437\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 2 } &= 268 + 134\cdot 457 + 199\cdot 457^{2} + 181\cdot 457^{3} + 337\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 3 } &= 331 + 223\cdot 457 + 448\cdot 457^{2} + 121\cdot 457^{3} + 278\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 4 } &= 361 + 161\cdot 457 + 409\cdot 457^{2} + 40\cdot 457^{3} + 119\cdot 457^{4} +O\left(457^{ 5 }\right) \\ r_{ 5 } &= 367 + 149\cdot 457 + 448\cdot 457^{2} + 7\cdot 457^{3} + 199\cdot 457^{4} +O\left(457^{ 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.