Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1931\) |
| Artin stem field: | Galois closure of 8.2.7200237491.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.1931.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1931.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 7x^{6} - 5x^{5} + x^{4} - 18x^{3} + 27x^{2} + 3x - 13 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9\cdot 13 + 7\cdot 13^{2} + 7\cdot 13^{3} + 2\cdot 13^{4} + 2\cdot 13^{5} + 2\cdot 13^{6} + 2\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 11 + \left(4 a + 5\right)\cdot 13 + \left(8 a + 6\right)\cdot 13^{2} + \left(4 a + 4\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(11 a + 12\right)\cdot 13^{5} + \left(4 a + 12\right)\cdot 13^{6} + \left(12 a + 10\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 10 + \left(2 a + 3\right)\cdot 13 + \left(2 a + 3\right)\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + 7\cdot 13^{4} + \left(8 a + 12\right)\cdot 13^{5} + \left(11 a + 12\right)\cdot 13^{6} + \left(7 a + 4\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 5 + \left(10 a + 11\right)\cdot 13 + \left(10 a + 2\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(12 a + 5\right)\cdot 13^{4} + \left(4 a + 7\right)\cdot 13^{5} + \left(a + 3\right)\cdot 13^{6} + \left(5 a + 1\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 4\cdot 13 + 6\cdot 13^{2} + 5\cdot 13^{3} + 10\cdot 13^{4} + 2\cdot 13^{5} + 13^{6} + 7\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 9 + \left(6 a + 1\right)\cdot 13 + 4\cdot 13^{2} + \left(7 a + 9\right)\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + 3 a\cdot 13^{5} + \left(9 a + 10\right)\cdot 13^{6} + \left(6 a + 4\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a + 8 + 8 a\cdot 13 + \left(4 a + 10\right)\cdot 13^{2} + 8 a\cdot 13^{3} + \left(a + 4\right)\cdot 13^{4} + a\cdot 13^{5} + \left(8 a + 6\right)\cdot 13^{6} + 5\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a + 2 + \left(6 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(5 a + 2\right)\cdot 13^{3} + \left(9 a + 4\right)\cdot 13^{4} + 9 a\cdot 13^{5} + \left(3 a + 3\right)\cdot 13^{6} + \left(6 a + 2\right)\cdot 13^{7} +O(13^{8})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,6)(3,5)(4,8)$ | $0$ |
| $8$ | $3$ | $(1,3,4)(5,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,3,5,6)(2,8,7,4)$ | $0$ |
| $8$ | $6$ | $(1,5)(2,8,6,7,4,3)$ | $1$ |
| $6$ | $8$ | $(1,4,6,7,5,8,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,8,6,2,5,4,3,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.