Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(415662001847\)\(\medspace = 17^{3} \cdot 439^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.7463.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.7463.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.7463.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + x^{3} - 2x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 25 + 11\cdot 41 + 12\cdot 41^{2} + 37\cdot 41^{3} + 34\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 8 + \left(14 a + 2\right)\cdot 41 + \left(38 a + 36\right)\cdot 41^{2} + \left(a + 38\right)\cdot 41^{3} + \left(5 a + 30\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 a + 29 + \left(26 a + 37\right)\cdot 41 + \left(2 a + 13\right)\cdot 41^{2} + \left(39 a + 6\right)\cdot 41^{3} + \left(35 a + 3\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 28 + \left(13 a + 23\right)\cdot 41 + \left(33 a + 7\right)\cdot 41^{2} + \left(35 a + 24\right)\cdot 41^{3} + \left(21 a + 32\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 35 + \left(27 a + 6\right)\cdot 41 + \left(7 a + 12\right)\cdot 41^{2} + \left(5 a + 16\right)\cdot 41^{3} + \left(19 a + 21\right)\cdot 41^{4} +O(41^{5})\)
|
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.