Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3267\)\(\medspace = 3^{3} \cdot 11^{2} \) |
| Artin stem field: | Galois closure of 8.2.32019867.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.3267.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 7x^{6} - 7x^{5} + 7x^{4} - 7x^{3} + 2x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 9 + \left(27 a + 9\right)\cdot 29 + \left(14 a + 27\right)\cdot 29^{2} + \left(7 a + 18\right)\cdot 29^{3} + \left(7 a + 1\right)\cdot 29^{4} + \left(22 a + 9\right)\cdot 29^{5} + \left(24 a + 2\right)\cdot 29^{6} + \left(5 a + 23\right)\cdot 29^{7} + \left(8 a + 4\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 4 + \left(a + 5\right)\cdot 29 + \left(14 a + 16\right)\cdot 29^{2} + \left(21 a + 12\right)\cdot 29^{3} + \left(21 a + 1\right)\cdot 29^{4} + \left(6 a + 26\right)\cdot 29^{5} + \left(4 a + 16\right)\cdot 29^{6} + \left(23 a + 27\right)\cdot 29^{7} + \left(20 a + 10\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 22\cdot 29 + 22\cdot 29^{2} + 28\cdot 29^{3} + 27\cdot 29^{4} + 24\cdot 29^{6} + 6\cdot 29^{7} + 13\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 21 + \left(a + 19\right)\cdot 29 + \left(14 a + 1\right)\cdot 29^{2} + \left(21 a + 10\right)\cdot 29^{3} + \left(21 a + 27\right)\cdot 29^{4} + \left(6 a + 19\right)\cdot 29^{5} + \left(4 a + 26\right)\cdot 29^{6} + \left(23 a + 5\right)\cdot 29^{7} + \left(20 a + 24\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 26 + \left(27 a + 23\right)\cdot 29 + \left(14 a + 12\right)\cdot 29^{2} + \left(7 a + 16\right)\cdot 29^{3} + \left(7 a + 27\right)\cdot 29^{4} + \left(22 a + 2\right)\cdot 29^{5} + \left(24 a + 12\right)\cdot 29^{6} + \left(5 a + 1\right)\cdot 29^{7} + \left(8 a + 18\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 + 6\cdot 29 + 6\cdot 29^{2} + 29^{4} + 28\cdot 29^{5} + 4\cdot 29^{6} + 22\cdot 29^{7} + 15\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a + 17 + \left(11 a + 3\right)\cdot 29 + \left(22 a + 22\right)\cdot 29^{2} + \left(27 a + 28\right)\cdot 29^{3} + 25\cdot 29^{4} + \left(8 a + 23\right)\cdot 29^{5} + \left(5 a + 19\right)\cdot 29^{6} + \left(a + 28\right)\cdot 29^{7} + \left(3 a + 21\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 24 a + 13 + \left(17 a + 25\right)\cdot 29 + \left(6 a + 6\right)\cdot 29^{2} + a\cdot 29^{3} + \left(28 a + 3\right)\cdot 29^{4} + \left(20 a + 5\right)\cdot 29^{5} + \left(23 a + 9\right)\cdot 29^{6} + 27 a\cdot 29^{7} + \left(25 a + 7\right)\cdot 29^{8} +O(29^{9})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,6)(3,4)(7,8)$ | $0$ |
| $8$ | $3$ | $(1,5,7)(2,8,4)$ | $-1$ |
| $6$ | $4$ | $(1,3,4,6)(2,8,5,7)$ | $0$ |
| $8$ | $6$ | $(1,8,5,4,7,2)(3,6)$ | $1$ |
| $6$ | $8$ | $(1,5,3,7,4,2,6,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,2,3,8,4,5,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.