Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(43875782047568\)\(\medspace = 2^{4} \cdot 13997^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.223952.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.13997.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.223952.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 6x^{3} - 2x^{2} + 6x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 146\cdot 173 + 116\cdot 173^{2} + 98\cdot 173^{3} + 84\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 48\cdot 173 + 14\cdot 173^{2} + 25\cdot 173^{3} + 164\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 58 + 145\cdot 173 + 138\cdot 173^{2} + 122\cdot 173^{3} + 44\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 100 + 94\cdot 173 + 35\cdot 173^{2} + 110\cdot 173^{3} + 49\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 109 + 84\cdot 173 + 40\cdot 173^{2} + 162\cdot 173^{3} + 2\cdot 173^{4} +O(173^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.