Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(5864\)\(\medspace = 2^{3} \cdot 733 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5864.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.5864.2t1.b.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.5864.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{3} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 16 + \left(35 a + 6\right)\cdot 47 + 41\cdot 47^{2} + \left(a + 24\right)\cdot 47^{3} + \left(8 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 18 + \left(25 a + 25\right)\cdot 47 + \left(33 a + 23\right)\cdot 47^{2} + \left(18 a + 7\right)\cdot 47^{3} + \left(37 a + 26\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a + 11 + \left(11 a + 9\right)\cdot 47 + \left(46 a + 7\right)\cdot 47^{2} + \left(45 a + 26\right)\cdot 47^{3} + \left(38 a + 34\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 13 + \left(21 a + 8\right)\cdot 47 + \left(13 a + 18\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(9 a + 35\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 44\cdot 47 + 3\cdot 47^{2} + 24\cdot 47^{3} + 25\cdot 47^{4} +O(47^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |