Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(351910769291801\)\(\medspace = 17^{3} \cdot 4153^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.70601.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.70601.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.70601.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 50\cdot 53 + 22\cdot 53^{2} + 49\cdot 53^{3} + 42\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 28 + \left(43 a + 17\right)\cdot 53 + \left(22 a + 45\right)\cdot 53^{2} + \left(11 a + 30\right)\cdot 53^{3} + \left(35 a + 13\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 a + 52 + \left(9 a + 24\right)\cdot 53 + \left(30 a + 40\right)\cdot 53^{2} + 41 a\cdot 53^{3} + \left(17 a + 37\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 18\cdot 53 + 50\cdot 53^{2} + 13\cdot 53^{3} + 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 47\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 11\cdot 53^{4} +O(53^{5})\)
|
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.