Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(20164\)\(\medspace = 2^{2} \cdot 71^{2} \) |
| Artin stem field: | Galois closure of 8.2.1626347584.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.20164.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - x^{6} - 4x^{5} + 6x^{4} - 4x^{3} - x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 14 + \left(9 a + 10\right)\cdot 19 + \left(5 a + 15\right)\cdot 19^{2} + \left(14 a + 6\right)\cdot 19^{3} + \left(6 a + 13\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(4 a + 12\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 12 + \left(a + 2\right)\cdot 19 + \left(6 a + 8\right)\cdot 19^{2} + \left(12 a + 15\right)\cdot 19^{3} + \left(8 a + 7\right)\cdot 19^{4} + \left(4 a + 18\right)\cdot 19^{5} + 9\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 3\cdot 19 + 18\cdot 19^{2} + 9\cdot 19^{3} + 5\cdot 19^{4} + 7\cdot 19^{5} + 10\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 1 + \left(17 a + 9\right)\cdot 19 + 18\cdot 19^{2} + \left(5 a + 3\right)\cdot 19^{3} + \left(7 a + 17\right)\cdot 19^{4} + 11 a\cdot 19^{5} + \left(4 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 9 + \left(17 a + 7\right)\cdot 19 + \left(12 a + 12\right)\cdot 19^{2} + \left(6 a + 2\right)\cdot 19^{3} + \left(10 a + 4\right)\cdot 19^{4} + \left(14 a + 14\right)\cdot 19^{5} + \left(18 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 + 5\cdot 19 + 9\cdot 19^{2} + 13\cdot 19^{3} + 2\cdot 19^{4} + 10\cdot 19^{5} + 9\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a + 15 + \left(a + 12\right)\cdot 19 + \left(18 a + 1\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + 11 a\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 4 a + 10 + \left(9 a + 5\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(4 a + 15\right)\cdot 19^{3} + \left(12 a + 5\right)\cdot 19^{4} + \left(7 a + 12\right)\cdot 19^{5} + \left(14 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $12$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,3,2)(6,7,8)$ | $-1$ | |
| $6$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | |
| $8$ | $6$ | $(1,8)(2,3,5,7,6,4)$ | $1$ | |
| $6$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ | |
| $6$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |