Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1099\)\(\medspace = 7 \cdot 157 \) |
| Artin stem field: | Galois closure of 8.2.1327373299.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.1099.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1099.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 2x^{6} - 3x^{4} + 11x^{3} - 13x^{2} + 15x - 11 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 10 + \left(3 a + 23\right)\cdot 29 + \left(27 a + 14\right)\cdot 29^{2} + \left(19 a + 11\right)\cdot 29^{3} + \left(28 a + 11\right)\cdot 29^{4} + \left(25 a + 23\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 29 + 5\cdot 29^{2} + 16\cdot 29^{3} + 2\cdot 29^{4} + 9\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 26 + \left(18 a + 18\right)\cdot 29 + \left(19 a + 4\right)\cdot 29^{2} + \left(15 a + 4\right)\cdot 29^{3} + \left(10 a + 4\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 10 + \left(24 a + 5\right)\cdot 29 + a\cdot 29^{2} + \left(18 a + 24\right)\cdot 29^{3} + \left(25 a + 8\right)\cdot 29^{4} + 22\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 26 + \left(4 a + 1\right)\cdot 29 + \left(27 a + 14\right)\cdot 29^{2} + \left(10 a + 25\right)\cdot 29^{3} + \left(3 a + 2\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 14 + \left(10 a + 9\right)\cdot 29 + \left(9 a + 26\right)\cdot 29^{2} + \left(13 a + 4\right)\cdot 29^{3} + \left(18 a + 12\right)\cdot 29^{4} + \left(7 a + 2\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 17 a + 12 + \left(25 a + 28\right)\cdot 29 + \left(a + 1\right)\cdot 29^{2} + \left(9 a + 26\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(3 a + 8\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 15 + 27\cdot 29 + 19\cdot 29^{2} + 3\cdot 29^{3} + 26\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $12$ | $2$ | $(1,4)(3,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(2,3,6)(4,8,5)$ | $-1$ |
| $6$ | $4$ | $(1,5,7,3)(2,4,8,6)$ | $0$ |
| $8$ | $6$ | $(1,5,8,7,3,2)(4,6)$ | $1$ |
| $6$ | $8$ | $(1,8,6,3,7,2,4,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,6,5,7,8,4,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.