Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2243\) |
| Artin stem field: | Galois closure of 8.2.11284642907.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.2243.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2243.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 9x^{5} + 11x^{4} + 35x^{3} + 4x^{2} - 74x - 85 \)
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The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 29\cdot 41 + 33\cdot 41^{2} + 23\cdot 41^{3} + 13\cdot 41^{4} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 a + 38 + \left(24 a + 29\right)\cdot 41 + \left(33 a + 27\right)\cdot 41^{2} + \left(38 a + 16\right)\cdot 41^{3} + \left(8 a + 24\right)\cdot 41^{4} + \left(35 a + 9\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 31 + \left(33 a + 33\right)\cdot 41 + \left(2 a + 12\right)\cdot 41^{2} + \left(24 a + 25\right)\cdot 41^{3} + \left(30 a + 36\right)\cdot 41^{4} + \left(14 a + 12\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 31\cdot 41 + 2\cdot 41^{2} + 34\cdot 41^{3} + 40\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 a + 2 + \left(7 a + 6\right)\cdot 41 + \left(38 a + 29\right)\cdot 41^{2} + \left(16 a + 12\right)\cdot 41^{3} + \left(10 a + 22\right)\cdot 41^{4} + \left(26 a + 26\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 26 + \left(16 a + 26\right)\cdot 41 + \left(7 a + 21\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(32 a + 12\right)\cdot 41^{4} + \left(5 a + 24\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 20 a + 32 + \left(15 a + 10\right)\cdot 41 + \left(32 a + 18\right)\cdot 41^{2} + \left(15 a + 9\right)\cdot 41^{3} + \left(39 a + 37\right)\cdot 41^{4} + \left(14 a + 1\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 a + 10 + \left(25 a + 37\right)\cdot 41 + \left(8 a + 17\right)\cdot 41^{2} + \left(25 a + 24\right)\cdot 41^{3} + \left(a + 16\right)\cdot 41^{4} + \left(26 a + 7\right)\cdot 41^{5} +O(41^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-2$ |
| $12$ | $2$ | $(2,6)(3,5)(7,8)$ | $0$ |
| $8$ | $3$ | $(1,6,3)(4,8,5)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,8)(2,5,7,3)$ | $0$ |
| $8$ | $6$ | $(1,5,6,4,3,8)(2,7)$ | $1$ |
| $6$ | $8$ | $(1,6,5,2,4,8,3,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,8,5,7,4,6,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.