Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4310084492032\)\(\medspace = 2^{8} \cdot 11^{3} \cdot 233^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.41008.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.10252.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.41008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{3} - 2x^{2} - 3x - 2 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 32\cdot 113 + 89\cdot 113^{2} + 68\cdot 113^{3} + 77\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 79\cdot 113 + 68\cdot 113^{2} + 72\cdot 113^{3} + 101\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 + 14\cdot 113 + 31\cdot 113^{2} + 38\cdot 113^{3} + 97\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 59 + 21\cdot 113 + 98\cdot 113^{2} + 113^{3} + 22\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 100 + 78\cdot 113 + 51\cdot 113^{2} + 44\cdot 113^{3} + 40\cdot 113^{4} +O(113^{5})\)
|
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.