# Properties

 Label 4.1944559427246513.10t12.a.a Dimension 4 Group $S_5$ Conductor $7^{3} \cdot 11^{3} \cdot 1621^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $1944559427246513= 7^{3} \cdot 11^{3} \cdot 1621^{3}$ Artin number field: Splitting field of 5.5.124817.1 defined by $f= x^{5} - 7 x^{3} - 6 x^{2} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.124817.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.5.124817.1

## Galois action

### Roots of defining polynomial

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

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