Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(491\) |
| Artin number field: | Galois closure of 8.2.118370771.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.491.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 37 a + 31 + \left(34 a + 5\right)\cdot 47 + \left(4 a + 25\right)\cdot 47^{2} + \left(27 a + 24\right)\cdot 47^{3} + \left(33 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 20 + \left(46 a + 32\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(6 a + 34\right)\cdot 47^{3} + \left(7 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 11 + \left(12 a + 38\right)\cdot 47 + \left(42 a + 46\right)\cdot 47^{2} + \left(19 a + 26\right)\cdot 47^{3} + \left(13 a + 35\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 a + 30 + 25\cdot 47 + \left(7 a + 24\right)\cdot 47^{2} + \left(40 a + 8\right)\cdot 47^{3} + \left(39 a + 3\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 44 + \left(19 a + 21\right)\cdot 47 + \left(21 a + 17\right)\cdot 47^{2} + \left(15 a + 45\right)\cdot 47^{3} + \left(3 a + 5\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 + 19\cdot 47 + 34\cdot 47^{2} + 23\cdot 47^{3} + 39\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 a + 13 + \left(27 a + 5\right)\cdot 47 + \left(25 a + 41\right)\cdot 47^{2} + \left(31 a + 7\right)\cdot 47^{3} + \left(43 a + 44\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 + 39\cdot 47 + 7\cdot 47^{2} + 16\cdot 47^{3} + 21\cdot 47^{4} +O(47^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(1,5)(3,8)(6,7)$ | $0$ | $0$ |
| $8$ | $3$ | $(1,4,8)(2,6,5)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,4,5,2)(3,6,7,8)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,6,4,5,8,2)(3,7)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,6,2,3,5,8,4,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,8,2,7,5,6,4,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |