Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(357306420736\)\(\medspace = 2^{9} \cdot 887^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.7096.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.7096.2t1.b.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.7096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 2 + \left(5 a + 8\right)\cdot 19 + \left(8 a + 8\right)\cdot 19^{2} + \left(18 a + 1\right)\cdot 19^{3} + \left(10 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 18 + \left(13 a + 16\right)\cdot 19 + \left(10 a + 10\right)\cdot 19^{2} + 11\cdot 19^{3} + \left(8 a + 4\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 13\cdot 19 + 17\cdot 19^{2} + 14\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 9 + \left(a + 1\right)\cdot 19 + \left(12 a + 14\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 14 + \left(17 a + 16\right)\cdot 19 + \left(6 a + 5\right)\cdot 19^{2} + \left(2 a + 7\right)\cdot 19^{3} + \left(17 a + 14\right)\cdot 19^{4} +O(19^{5})\)
|
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.