Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(7544528179397\)\(\medspace = 11^{3} \cdot 1783^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.19613.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.19613.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.19613.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 4x^{3} - 6x^{2} + 3x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 15 a + 5 + \left(2 a + 12\right)\cdot 19 + \left(16 a + 4\right)\cdot 19^{2} + \left(16 a + 18\right)\cdot 19^{3} + \left(5 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 1 + 16 a\cdot 19 + \left(2 a + 18\right)\cdot 19^{2} + \left(2 a + 18\right)\cdot 19^{3} + \left(13 a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 9 + \left(6 a + 14\right)\cdot 19 + \left(11 a + 1\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(10 a + 2\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 15\cdot 19 + 6\cdot 19^{2} + 10\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 15 + \left(12 a + 14\right)\cdot 19 + \left(7 a + 6\right)\cdot 19^{2} + \left(2 a + 7\right)\cdot 19^{3} + \left(8 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.