Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(643\) |
| Artin stem field: | Galois closure of 8.2.265847707.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.643.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.643.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 3x^{6} + x^{5} - 7x^{4} + 18x^{3} - 12x^{2} + 4x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + \left(19 a + 6\right)\cdot 37 + \left(14 a + 34\right)\cdot 37^{2} + \left(2 a + 12\right)\cdot 37^{3} + \left(12 a + 36\right)\cdot 37^{4} + \left(5 a + 4\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 15 + \left(17 a + 33\right)\cdot 37 + \left(22 a + 35\right)\cdot 37^{2} + \left(34 a + 7\right)\cdot 37^{3} + \left(24 a + 8\right)\cdot 37^{4} + \left(31 a + 14\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 4 + \left(4 a + 30\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(28 a + 27\right)\cdot 37^{3} + \left(19 a + 33\right)\cdot 37^{4} + \left(8 a + 2\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 8 + 22 a\cdot 37 + \left(23 a + 21\right)\cdot 37^{2} + \left(20 a + 24\right)\cdot 37^{3} + \left(30 a + 28\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 + 37^{2} + 23\cdot 37^{3} + 19\cdot 37^{4} + 14\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a + 11 + \left(32 a + 36\right)\cdot 37 + \left(35 a + 7\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(17 a + 10\right)\cdot 37^{4} + \left(28 a + 17\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 11 + 31\cdot 37 + 20\cdot 37^{2} + 15\cdot 37^{3} + 28\cdot 37^{4} + 4\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 a + 28 + \left(14 a + 9\right)\cdot 37 + \left(13 a + 19\right)\cdot 37^{2} + \left(16 a + 9\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} + \left(2 a + 6\right)\cdot 37^{5} +O(37^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,2)(5,8)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,8,3)(2,6,4)$ | $-1$ |
| $6$ | $4$ | $(1,7,2,5)(3,6,4,8)$ | $0$ |
| $8$ | $6$ | $(1,4,8,2,3,6)(5,7)$ | $1$ |
| $6$ | $8$ | $(1,5,3,6,2,7,4,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,7,3,8,2,5,4,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.