# Properties

 Label 4.11_523.5t5.1c1 Dimension 4 Group $S_5$ Conductor $11 \cdot 523$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $5753= 11 \cdot 523$ Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} + 2 x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.11_523.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $103 + 452\cdot 467 + 304\cdot 467^{2} + 160\cdot 467^{3} + 237\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 2 }$ $=$ $209 + 205\cdot 467 + 99\cdot 467^{2} + 299\cdot 467^{3} + 120\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 3 }$ $=$ $256 + 414\cdot 467 + 251\cdot 467^{2} + 329\cdot 467^{3} + 236\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 4 }$ $=$ $397 + 338\cdot 467 + 360\cdot 467^{2} + 288\cdot 467^{3} + 198\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 5 }$ $=$ $437 + 456\cdot 467 + 383\cdot 467^{2} + 322\cdot 467^{3} + 140\cdot 467^{4} +O\left(467^{ 5 }\right)$

### 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.