Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(13534019565191\)\(\medspace = 23831^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.23831.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.23831.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.23831.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 3x^{3} - 2x^{2} + 4x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 5\cdot 223 + 175\cdot 223^{2} + 199\cdot 223^{3} + 120\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 50\cdot 223 + 35\cdot 223^{2} + 203\cdot 223^{3} + 128\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 76 + 139\cdot 223 + 124\cdot 223^{2} + 74\cdot 223^{3} + 184\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 125 + 36\cdot 223 + 172\cdot 223^{2} + 102\cdot 223^{3} + 7\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 173 + 214\cdot 223 + 161\cdot 223^{2} + 88\cdot 223^{3} + 4\cdot 223^{4} +O(223^{5})\)
|
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.