Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(3233\)\(\medspace = 53 \cdot 61 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.3233.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.3233.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.3233.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{2} - 1 \)
|
The roots of $f$ are computed in $\Q_{ 383 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 88 + 282\cdot 383 + 41\cdot 383^{2} + 170\cdot 383^{3} + 220\cdot 383^{4} +O(383^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 89 + 155\cdot 383 + 93\cdot 383^{2} + 241\cdot 383^{3} + 206\cdot 383^{4} +O(383^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 120 + 178\cdot 383 + 256\cdot 383^{2} + 144\cdot 383^{3} + 377\cdot 383^{4} +O(383^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 127 + 109\cdot 383 + 48\cdot 383^{2} + 292\cdot 383^{3} + 181\cdot 383^{4} +O(383^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 342 + 40\cdot 383 + 326\cdot 383^{2} + 300\cdot 383^{3} + 162\cdot 383^{4} +O(383^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
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.