Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1048\)\(\medspace = 2^{3} \cdot 131 \) |
| Artin stem field: | Galois closure of 8.2.2302045184.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.131.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} - 4x^{5} + 2x^{4} - 6x^{3} - 4x^{2} - 18x - 29 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 8\cdot 29 + 27\cdot 29^{3} + 22\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 1 + \left(24 a + 19\right)\cdot 29 + 4\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(20 a + 1\right)\cdot 29^{4} + \left(6 a + 20\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + \left(7 a + 14\right)\cdot 29 + \left(9 a + 9\right)\cdot 29^{2} + \left(3 a + 23\right)\cdot 29^{3} + \left(21 a + 26\right)\cdot 29^{4} + \left(26 a + 9\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 4 + 21 a\cdot 29 + \left(19 a + 19\right)\cdot 29^{2} + \left(25 a + 1\right)\cdot 29^{3} + \left(7 a + 13\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 22\cdot 29 + 18\cdot 29^{2} + 19\cdot 29^{3} + 7\cdot 29^{4} + 25\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 a + 3 + \left(3 a + 20\right)\cdot 29 + \left(4 a + 16\right)\cdot 29^{2} + \left(2 a + 5\right)\cdot 29^{3} + \left(17 a + 21\right)\cdot 29^{4} + \left(23 a + 16\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 a + 16 + \left(4 a + 20\right)\cdot 29 + \left(28 a + 13\right)\cdot 29^{2} + \left(18 a + 23\right)\cdot 29^{3} + \left(8 a + 5\right)\cdot 29^{4} + \left(22 a + 4\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( a + 27 + \left(25 a + 11\right)\cdot 29 + \left(24 a + 4\right)\cdot 29^{2} + \left(26 a + 12\right)\cdot 29^{3} + \left(11 a + 17\right)\cdot 29^{4} + \left(5 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(3,8)(4,6)$ | $0$ |
| $8$ | $3$ | $(1,4,3)(5,8,6)$ | $-1$ |
| $6$ | $4$ | $(1,3,5,6)(2,4,7,8)$ | $0$ |
| $8$ | $6$ | $(1,8,3,5,4,6)(2,7)$ | $1$ |
| $6$ | $8$ | $(1,6,7,8,5,3,2,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,3,7,4,5,6,2,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.