Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(981\)\(\medspace = 3^{2} \cdot 109 \) |
| Artin stem field: | Galois closure of 8.2.2832228423.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.327.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2943.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 10x^{4} + 2x^{3} + 10x^{2} - 13x - 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 4 + 21\cdot 29 + \left(4 a + 4\right)\cdot 29^{2} + \left(3 a + 23\right)\cdot 29^{3} + \left(28 a + 3\right)\cdot 29^{4} + \left(3 a + 4\right)\cdot 29^{5} + \left(15 a + 22\right)\cdot 29^{6} + \left(19 a + 16\right)\cdot 29^{7} + \left(24 a + 20\right)\cdot 29^{8} + \left(25 a + 5\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 23 + \left(21 a + 13\right)\cdot 29 + \left(12 a + 28\right)\cdot 29^{2} + \left(23 a + 13\right)\cdot 29^{3} + \left(15 a + 10\right)\cdot 29^{4} + \left(10 a + 20\right)\cdot 29^{5} + \left(26 a + 26\right)\cdot 29^{6} + \left(11 a + 22\right)\cdot 29^{7} + \left(20 a + 16\right)\cdot 29^{8} + \left(12 a + 21\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 14 + \left(21 a + 10\right)\cdot 29 + \left(12 a + 16\right)\cdot 29^{2} + \left(23 a + 26\right)\cdot 29^{3} + \left(15 a + 20\right)\cdot 29^{4} + 10 a\cdot 29^{5} + \left(26 a + 26\right)\cdot 29^{6} + \left(11 a + 1\right)\cdot 29^{7} + \left(20 a + 9\right)\cdot 29^{8} + \left(12 a + 22\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 26 + \left(28 a + 7\right)\cdot 29 + \left(24 a + 24\right)\cdot 29^{2} + \left(25 a + 5\right)\cdot 29^{3} + 25\cdot 29^{4} + \left(25 a + 24\right)\cdot 29^{5} + \left(13 a + 6\right)\cdot 29^{6} + \left(9 a + 12\right)\cdot 29^{7} + \left(4 a + 8\right)\cdot 29^{8} + \left(3 a + 23\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 18\cdot 29 + 27\cdot 29^{2} + 29^{3} + 2\cdot 29^{4} + 23\cdot 29^{6} + 20\cdot 29^{7} + 6\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 7 + \left(7 a + 15\right)\cdot 29 + 16 a\cdot 29^{2} + \left(5 a + 15\right)\cdot 29^{3} + \left(13 a + 18\right)\cdot 29^{4} + \left(18 a + 8\right)\cdot 29^{5} + \left(2 a + 2\right)\cdot 29^{6} + \left(17 a + 6\right)\cdot 29^{7} + \left(8 a + 12\right)\cdot 29^{8} + \left(16 a + 7\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 + 10\cdot 29 + 29^{2} + 27\cdot 29^{3} + 26\cdot 29^{4} + 28\cdot 29^{5} + 5\cdot 29^{6} + 8\cdot 29^{7} + 22\cdot 29^{8} + 16\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 13 a + 16 + \left(7 a + 18\right)\cdot 29 + \left(16 a + 12\right)\cdot 29^{2} + \left(5 a + 2\right)\cdot 29^{3} + \left(13 a + 8\right)\cdot 29^{4} + \left(18 a + 28\right)\cdot 29^{5} + \left(2 a + 2\right)\cdot 29^{6} + \left(17 a + 27\right)\cdot 29^{7} + \left(8 a + 19\right)\cdot 29^{8} + \left(16 a + 6\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,8)(3,6)$ | $0$ |
| $8$ | $3$ | $(1,2,5)(4,6,7)$ | $-1$ |
| $6$ | $4$ | $(1,3,4,8)(2,5,6,7)$ | $0$ |
| $8$ | $6$ | $(1,5,3,4,7,8)(2,6)$ | $1$ |
| $6$ | $8$ | $(1,8,5,6,4,3,7,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,3,5,2,4,8,7,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.