Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1423\) |
| Artin stem field: | Galois closure of 8.2.2881473967.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.1423.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1423.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + x^{6} + 7x^{5} - 13x^{4} - 4x^{3} + 23x^{2} - 25x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 a + 25 + \left(15 a + 13\right)\cdot 31 + \left(15 a + 14\right)\cdot 31^{2} + \left(21 a + 10\right)\cdot 31^{3} + \left(29 a + 25\right)\cdot 31^{4} + \left(18 a + 16\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 24 + \left(9 a + 20\right)\cdot 31 + \left(7 a + 27\right)\cdot 31^{2} + \left(12 a + 29\right)\cdot 31^{3} + \left(8 a + 18\right)\cdot 31^{4} + \left(14 a + 1\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 26\cdot 31 + 2\cdot 31^{2} + 27\cdot 31^{3} + 22\cdot 31^{4} + 27\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 12 + 7\cdot 31 + \left(3 a + 29\right)\cdot 31^{2} + \left(18 a + 11\right)\cdot 31^{3} + \left(19 a + 7\right)\cdot 31^{4} + \left(15 a + 10\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 17 + \left(15 a + 18\right)\cdot 31 + \left(15 a + 29\right)\cdot 31^{2} + \left(9 a + 6\right)\cdot 31^{3} + \left(a + 1\right)\cdot 31^{4} + \left(12 a + 25\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 + 7\cdot 31 + 14\cdot 31^{2} + 7\cdot 31^{3} + 27\cdot 31^{4} + 29\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 a + 28 + \left(30 a + 30\right)\cdot 31 + \left(27 a + 3\right)\cdot 31^{2} + \left(12 a + 14\right)\cdot 31^{3} + \left(11 a + 28\right)\cdot 31^{4} + \left(15 a + 21\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 a + 13 + \left(21 a + 29\right)\cdot 31 + \left(23 a + 1\right)\cdot 31^{2} + \left(18 a + 16\right)\cdot 31^{3} + \left(22 a + 23\right)\cdot 31^{4} + \left(16 a + 21\right)\cdot 31^{5} +O(31^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $-2$ |
| $12$ | $2$ | $(1,6)(3,8)(4,7)$ | $0$ |
| $8$ | $3$ | $(2,6,7)(3,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,4,8,7)(2,6,5,3)$ | $0$ |
| $8$ | $6$ | $(1,8)(2,4,6,5,7,3)$ | $1$ |
| $6$ | $8$ | $(1,5,3,7,8,2,6,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,2,3,4,8,5,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.