Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4417\)\(\medspace = 7 \cdot 631 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.4417.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.4417.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.4417.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 2x^{3} - x^{2} - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 11 a + 18 + \left(31 a + 26\right)\cdot 41 + \left(4 a + 3\right)\cdot 41^{2} + \left(13 a + 17\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 a + 10 + \left(9 a + 27\right)\cdot 41 + \left(36 a + 27\right)\cdot 41^{2} + \left(27 a + 10\right)\cdot 41^{3} + \left(33 a + 31\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 35\cdot 41 + 40\cdot 41^{2} + 4\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 33\cdot 41 + 13\cdot 41^{2} + 10\cdot 41^{3} + 13\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 40\cdot 41 + 36\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} +O(41^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |