Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(9334886729678693\)\(\medspace = 210557^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.210557.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.210557.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.210557.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 6x^{3} + 4x^{2} + 8x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 29\cdot 43 + 10\cdot 43^{2} + 37\cdot 43^{3} + 40\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 41 + 14 a\cdot 43 + \left(11 a + 5\right)\cdot 43^{2} + \left(12 a + 5\right)\cdot 43^{3} + \left(11 a + 8\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 14 + \left(28 a + 42\right)\cdot 43 + \left(31 a + 1\right)\cdot 43^{2} + \left(30 a + 6\right)\cdot 43^{3} + \left(31 a + 7\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 24\cdot 43 + 29\cdot 43^{2} + 2\cdot 43^{3} + 39\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 32\cdot 43 + 38\cdot 43^{2} + 34\cdot 43^{3} + 33\cdot 43^{4} +O(43^{5})\)
|
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.