Basic invariants
| Dimension: | $6$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(14974189568\)\(\medspace = 2^{19} \cdot 13^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.58492928.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 6.0.58492928.4 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 3x^{4} - 6x^{3} - x^{2} + 23x + 27 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 17 a + 15 + \left(8 a + 11\right)\cdot 19 + \left(17 a + 11\right)\cdot 19^{2} + \left(7 a + 9\right)\cdot 19^{3} + \left(13 a + 7\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 13 + \left(10 a + 3\right)\cdot 19 + \left(a + 1\right)\cdot 19^{2} + 11 a\cdot 19^{3} + \left(5 a + 13\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 6\cdot 19 + 2\cdot 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 19 + 14\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 13 + \left(3 a + 3\right)\cdot 19 + \left(11 a + 17\right)\cdot 19^{2} + \left(11 a + 3\right)\cdot 19^{3} + \left(2 a + 6\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 10 + \left(15 a + 10\right)\cdot 19 + \left(7 a + 5\right)\cdot 19^{2} + \left(7 a + 4\right)\cdot 19^{3} + \left(16 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $6$ | |
| $10$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ | ✓ |
| $15$ | $2$ | $(1,2)(5,6)$ | $-2$ | |
| $20$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ | |
| $30$ | $4$ | $(1,6,4,2)$ | $0$ | |
| $24$ | $5$ | $(1,4,5,6,3)$ | $1$ | |
| $20$ | $6$ | $(1,6,2,3,5,4)$ | $0$ |