Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(11356184145377917\)\(\medspace = 53^{3} \cdot 4241^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.224773.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | even |
| Determinant: | 1.224773.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.224773.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 7x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 6\cdot 23 + 23^{2} + 19\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 2 + \left(6 a + 7\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{3} + \left(4 a + 19\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 7 + \left(16 a + 6\right)\cdot 23 + \left(22 a + 17\right)\cdot 23^{2} + \left(12 a + 7\right)\cdot 23^{3} + \left(18 a + 18\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 17\cdot 23 + 3\cdot 23^{2} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 8\cdot 23 + 8\cdot 23^{3} + 9\cdot 23^{4} +O(23^{5})\)
|
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.