Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1399\) |
| Artin stem field: | Galois closure of 8.2.2738124199.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.1399.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1399.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 10x^{5} + 4x^{4} + 4x^{3} - 14x^{2} + 11x - 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 21\cdot 47 + 11\cdot 47^{2} + 22\cdot 47^{3} + 43\cdot 47^{4} + 39\cdot 47^{5} + 11\cdot 47^{6} + 23\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 35 + 4 a\cdot 47 + \left(6 a + 8\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(2 a + 26\right)\cdot 47^{4} + \left(19 a + 30\right)\cdot 47^{5} + \left(34 a + 40\right)\cdot 47^{6} + \left(28 a + 1\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 25\cdot 47 + 35\cdot 47^{2} + 24\cdot 47^{3} + 3\cdot 47^{4} + 7\cdot 47^{5} + 35\cdot 47^{6} + 23\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 22 + \left(26 a + 45\right)\cdot 47 + \left(13 a + 22\right)\cdot 47^{2} + \left(42 a + 11\right)\cdot 47^{3} + 20\cdot 47^{4} + \left(41 a + 6\right)\cdot 47^{5} + \left(14 a + 29\right)\cdot 47^{6} + \left(42 a + 35\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 4 + \left(4 a + 18\right)\cdot 47 + \left(6 a + 31\right)\cdot 47^{2} + \left(38 a + 17\right)\cdot 47^{3} + \left(2 a + 6\right)\cdot 47^{4} + \left(19 a + 28\right)\cdot 47^{5} + \left(34 a + 3\right)\cdot 47^{6} + \left(28 a + 22\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a + 13 + \left(42 a + 46\right)\cdot 47 + \left(40 a + 38\right)\cdot 47^{2} + \left(8 a + 40\right)\cdot 47^{3} + \left(44 a + 20\right)\cdot 47^{4} + \left(27 a + 16\right)\cdot 47^{5} + \left(12 a + 6\right)\cdot 47^{6} + \left(18 a + 45\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 45 a + 26 + \left(20 a + 1\right)\cdot 47 + \left(33 a + 24\right)\cdot 47^{2} + \left(4 a + 35\right)\cdot 47^{3} + \left(46 a + 26\right)\cdot 47^{4} + \left(5 a + 40\right)\cdot 47^{5} + \left(32 a + 17\right)\cdot 47^{6} + \left(4 a + 11\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 19 a + 44 + \left(42 a + 28\right)\cdot 47 + \left(40 a + 15\right)\cdot 47^{2} + \left(8 a + 29\right)\cdot 47^{3} + \left(44 a + 40\right)\cdot 47^{4} + \left(27 a + 18\right)\cdot 47^{5} + \left(12 a + 43\right)\cdot 47^{6} + \left(18 a + 24\right)\cdot 47^{7} +O(47^{8})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $-2$ |
| $12$ | $2$ | $(2,7)(4,6)(5,8)$ | $0$ |
| $8$ | $3$ | $(2,4,5)(6,7,8)$ | $-1$ |
| $6$ | $4$ | $(1,4,3,7)(2,8,6,5)$ | $0$ |
| $8$ | $6$ | $(1,4,2,3,7,6)(5,8)$ | $1$ |
| $6$ | $8$ | $(1,6,8,4,3,2,5,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,8,7,3,6,5,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.