Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(541522735555229\)\(\medspace = 81509^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.81509.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.81509.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.81509.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 21 + 25\cdot 47 + 5\cdot 47^{2} + 25\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 16 + \left(35 a + 1\right)\cdot 47 + \left(39 a + 28\right)\cdot 47^{2} + \left(35 a + 22\right)\cdot 47^{3} + \left(8 a + 2\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 5\cdot 47 + 33\cdot 47^{2} + 9\cdot 47^{3} + 29\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 39\cdot 47 + 47^{2} + 29\cdot 47^{3} + 24\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 a + 20 + \left(11 a + 22\right)\cdot 47 + \left(7 a + 25\right)\cdot 47^{2} + \left(11 a + 7\right)\cdot 47^{3} + \left(38 a + 31\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.