Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3204\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 89 \) |
| Artin stem field: | Galois closure of 8.2.24668275248.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.267.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.9612.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 10x^{6} - 7x^{5} + 16x^{4} - 13x^{3} - 23x^{2} + 74x + 16 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 3 + \left(2 a + 8\right)\cdot 11 + 7 a\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(7 a + 2\right)\cdot 11^{4} + \left(2 a + 9\right)\cdot 11^{5} + \left(3 a + 6\right)\cdot 11^{6} + \left(a + 6\right)\cdot 11^{7} + \left(3 a + 6\right)\cdot 11^{8} +O(11^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 7 + \left(6 a + 1\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(6 a + 3\right)\cdot 11^{3} + \left(8 a + 8\right)\cdot 11^{4} + \left(9 a + 10\right)\cdot 11^{5} + \left(7 a + 9\right)\cdot 11^{6} + \left(9 a + 6\right)\cdot 11^{7} + 2 a\cdot 11^{8} +O(11^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 2\cdot 11^{2} + 4\cdot 11^{3} + 7\cdot 11^{4} + 2\cdot 11^{5} + 9\cdot 11^{6} + 7\cdot 11^{7} +O(11^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 9 + \left(2 a + 1\right)\cdot 11 + \left(2 a + 9\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} + \left(2 a + 7\right)\cdot 11^{4} + 10\cdot 11^{5} + \left(6 a + 3\right)\cdot 11^{6} + \left(10 a + 3\right)\cdot 11^{7} + 11^{8} +O(11^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 6 + \left(4 a + 9\right)\cdot 11 + \left(3 a + 3\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(2 a + 3\right)\cdot 11^{4} + \left(a + 8\right)\cdot 11^{5} + \left(3 a + 9\right)\cdot 11^{6} + \left(a + 4\right)\cdot 11^{7} + \left(8 a + 2\right)\cdot 11^{8} +O(11^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 10 + \left(8 a + 9\right)\cdot 11 + \left(3 a + 4\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + \left(3 a + 3\right)\cdot 11^{4} + \left(8 a + 1\right)\cdot 11^{5} + \left(7 a + 6\right)\cdot 11^{6} + \left(9 a + 8\right)\cdot 11^{7} + \left(7 a + 6\right)\cdot 11^{8} +O(11^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 + 5\cdot 11 + 5\cdot 11^{2} + 10\cdot 11^{3} + 9\cdot 11^{4} + 2\cdot 11^{5} + 3\cdot 11^{6} + 10\cdot 11^{7} + 8\cdot 11^{8} +O(11^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a + 7 + \left(8 a + 6\right)\cdot 11 + \left(8 a + 4\right)\cdot 11^{2} + \left(6 a + 10\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} + \left(10 a + 9\right)\cdot 11^{5} + \left(4 a + 5\right)\cdot 11^{6} + 6\cdot 11^{7} + \left(10 a + 5\right)\cdot 11^{8} +O(11^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ | |
| $12$ | $2$ | $(3,5)(4,8)(6,7)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,4,3)(2,8,7)$ | $-1$ | |
| $6$ | $4$ | $(1,7,2,3)(4,6,8,5)$ | $0$ | |
| $8$ | $6$ | $(1,7,4,2,3,8)(5,6)$ | $1$ | |
| $6$ | $8$ | $(1,7,8,6,2,3,4,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ | |
| $6$ | $8$ | $(1,3,8,5,2,7,4,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |