Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(18463\)\(\medspace = 37 \cdot 499 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.18463.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.18463.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.18463.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 3x^{3} + 4x^{2} + x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 32\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 27\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 2\cdot 61 + 22\cdot 61^{2} + 4\cdot 61^{3} + 46\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 29\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 17\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 8\cdot 61 + 32\cdot 61^{2} + 12\cdot 61^{3} + 6\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 + 48\cdot 61 + 14\cdot 61^{2} + 50\cdot 61^{3} + 24\cdot 61^{4} +O(61^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | ✓ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |