Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4182697396374561\)\(\medspace = 3^{3} \cdot 43^{3} \cdot 1249^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.161121.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.161121.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.161121.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 5 + \left(3 a + 23\right)\cdot 41 + \left(39 a + 5\right)\cdot 41^{2} + \left(24 a + 24\right)\cdot 41^{3} + \left(9 a + 40\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 a + 11 + \left(37 a + 30\right)\cdot 41 + \left(a + 37\right)\cdot 41^{2} + \left(16 a + 18\right)\cdot 41^{3} + \left(31 a + 3\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 18 + \left(17 a + 28\right)\cdot 41 + \left(28 a + 1\right)\cdot 41^{2} + \left(35 a + 40\right)\cdot 41^{3} + \left(2 a + 17\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 20\cdot 41 + 8\cdot 41^{2} + 3\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 34 + \left(23 a + 20\right)\cdot 41 + \left(12 a + 28\right)\cdot 41^{2} + \left(5 a + 36\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} +O(41^{5})\)
|
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.