Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(11135879290691\)\(\medspace = 137^{3} \cdot 163^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.22331.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | odd |
| Determinant: | 1.22331.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.22331.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 4x^{2} - x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 401 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 121 + 389\cdot 401 + 207\cdot 401^{2} + 292\cdot 401^{3} + 341\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 124 + 372\cdot 401 + 344\cdot 401^{2} + 74\cdot 401^{3} + 397\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 129 + 144\cdot 401 + 132\cdot 401^{2} + 193\cdot 401^{3} + 60\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 183 + 354\cdot 401 + 218\cdot 401^{2} + 41\cdot 401^{3} + 71\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 247 + 343\cdot 401 + 298\cdot 401^{2} + 199\cdot 401^{3} + 332\cdot 401^{4} +O(401^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.