Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(688\)\(\medspace = 2^{4} \cdot 43 \) |
| Artin stem field: | Galois closure of 8.2.325660672.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.43.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.688.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{6} - 12x^{5} + 16x^{4} - 18x^{3} + 10x^{2} - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 1 + \left(a + 15\right)\cdot 19 + \left(3 a + 4\right)\cdot 19^{2} + \left(2 a + 15\right)\cdot 19^{3} + 13 a\cdot 19^{4} + \left(6 a + 4\right)\cdot 19^{5} + \left(13 a + 13\right)\cdot 19^{6} +O(19^{7})\)
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| $r_{ 2 }$ | $=$ |
\( 8 + 10\cdot 19 + 5\cdot 19^{2} + 18\cdot 19^{3} + 16\cdot 19^{4} + 18\cdot 19^{5} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 16 + 5\cdot 19 + \left(2 a + 8\right)\cdot 19^{2} + \left(11 a + 5\right)\cdot 19^{3} + \left(3 a + 13\right)\cdot 19^{4} + \left(9 a + 6\right)\cdot 19^{5} + \left(2 a + 15\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 11 + \left(18 a + 11\right)\cdot 19 + \left(16 a + 9\right)\cdot 19^{2} + \left(7 a + 14\right)\cdot 19^{3} + \left(15 a + 5\right)\cdot 19^{4} + \left(9 a + 12\right)\cdot 19^{5} + \left(16 a + 8\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 2 + \left(13 a + 15\right)\cdot 19 + \left(7 a + 14\right)\cdot 19^{2} + \left(2 a + 5\right)\cdot 19^{3} + \left(8 a + 3\right)\cdot 19^{4} + \left(12 a + 6\right)\cdot 19^{5} + \left(6 a + 11\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 8 + \left(17 a + 9\right)\cdot 19 + \left(15 a + 6\right)\cdot 19^{2} + \left(16 a + 14\right)\cdot 19^{3} + \left(5 a + 11\right)\cdot 19^{4} + \left(12 a + 16\right)\cdot 19^{5} + 5 a\cdot 19^{6} +O(19^{7})\)
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| $r_{ 7 }$ | $=$ |
\( 18 a + 3 + \left(5 a + 8\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + 16 a\cdot 19^{3} + \left(10 a + 9\right)\cdot 19^{4} + \left(6 a + 10\right)\cdot 19^{5} + \left(12 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 + 17\cdot 19^{2} + 19^{3} + 15\cdot 19^{4} + 19^{6} +O(19^{7})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $12$ | $2$ | $(1,7)(2,8)(5,6)$ | $0$ |
| $8$ | $3$ | $(2,4,6)(3,7,8)$ | $-1$ |
| $6$ | $4$ | $(1,6,5,7)(2,4,8,3)$ | $0$ |
| $8$ | $6$ | $(1,6,4,5,7,3)(2,8)$ | $1$ |
| $6$ | $8$ | $(1,8,7,4,5,2,6,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,7,3,5,8,6,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.