Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(331\) |
| Artin stem field: | Galois closure of 8.2.36264691.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.331.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.331.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 5x^{6} - x^{5} - 3x^{4} + 3x^{3} - x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 12 + \left(9 a + 4\right)\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + 11 a\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(8 a + 5\right)\cdot 13^{6} + \left(8 a + 11\right)\cdot 13^{7} + 10\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\)
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| $r_{ 2 }$ | $=$ |
\( 2 a + 6 + \left(7 a + 10\right)\cdot 13 + \left(a + 2\right)\cdot 13^{2} + \left(2 a + 6\right)\cdot 13^{3} + \left(5 a + 11\right)\cdot 13^{4} + 8\cdot 13^{5} + \left(4 a + 4\right)\cdot 13^{6} + \left(a + 1\right)\cdot 13^{7} + \left(2 a + 6\right)\cdot 13^{8} + \left(4 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 10 + \left(3 a + 3\right)\cdot 13 + 4\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(10 a + 12\right)\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(4 a + 4\right)\cdot 13^{6} + \left(4 a + 11\right)\cdot 13^{7} + \left(12 a + 2\right)\cdot 13^{8} + \left(12 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 4 + \left(9 a + 9\right)\cdot 13 + \left(12 a + 8\right)\cdot 13^{2} + 11 a\cdot 13^{3} + 2 a\cdot 13^{4} + \left(10 a + 11\right)\cdot 13^{5} + \left(8 a + 8\right)\cdot 13^{6} + \left(8 a + 1\right)\cdot 13^{7} + 10\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\)
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| $r_{ 5 }$ | $=$ |
\( 9 + 13 + 9\cdot 13^{2} + 8\cdot 13^{3} + 4\cdot 13^{4} + 12\cdot 13^{5} + 12\cdot 13^{6} + 13^{7} + 5\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 2 + \left(3 a + 8\right)\cdot 13 + 11\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(10 a + 3\right)\cdot 13^{4} + \left(2 a + 5\right)\cdot 13^{5} + \left(4 a + 7\right)\cdot 13^{6} + \left(4 a + 1\right)\cdot 13^{7} + \left(12 a + 2\right)\cdot 13^{8} + \left(12 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 + 11\cdot 13 + 3\cdot 13^{2} + 4\cdot 13^{3} + 8\cdot 13^{4} + 11\cdot 13^{7} + 12\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 11 a + 8 + \left(5 a + 2\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(7 a + 1\right)\cdot 13^{4} + \left(12 a + 4\right)\cdot 13^{5} + \left(8 a + 8\right)\cdot 13^{6} + \left(11 a + 11\right)\cdot 13^{7} + \left(10 a + 6\right)\cdot 13^{8} + \left(8 a + 7\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $12$ | $2$ | $(1,3)(2,8)(4,6)$ | $0$ |
| $8$ | $3$ | $(1,5,8)(2,6,7)$ | $-1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5,8,7)$ | $0$ |
| $8$ | $6$ | $(1,2,5,6,8,7)(3,4)$ | $1$ |
| $6$ | $8$ | $(1,7,4,2,6,5,3,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,4,8,6,7,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.