Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2183\)\(\medspace = 37 \cdot 59 \) |
| Artin stem field: | Galois closure of 8.2.10403062487.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.2183.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2183.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 5x^{6} - x^{5} + 12x^{4} + 18x^{3} - 15x^{2} - 27x + 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 36 + 57\cdot 79 + 33\cdot 79^{2} + 40\cdot 79^{3} + 42\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 51 + \left(a + 2\right)\cdot 79 + \left(7 a + 39\right)\cdot 79^{2} + \left(a + 38\right)\cdot 79^{3} + \left(12 a + 52\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 45\cdot 79 + 27\cdot 79^{2} + 50\cdot 79^{3} + 57\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 a + 35 + \left(77 a + 27\right)\cdot 79 + \left(34 a + 4\right)\cdot 79^{2} + \left(47 a + 50\right)\cdot 79^{3} + \left(76 a + 13\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 54 a + 59 + \left(55 a + 70\right)\cdot 79 + \left(73 a + 53\right)\cdot 79^{2} + \left(33 a + 40\right)\cdot 79^{3} + \left(41 a + 57\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 76 a + 54 + 77 a\cdot 79 + \left(71 a + 45\right)\cdot 79^{2} + \left(77 a + 32\right)\cdot 79^{3} + \left(66 a + 63\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a + 22 + \left(a + 39\right)\cdot 79 + \left(44 a + 40\right)\cdot 79^{2} + \left(31 a + 62\right)\cdot 79^{3} + \left(2 a + 42\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 a + 34 + \left(23 a + 72\right)\cdot 79 + \left(5 a + 71\right)\cdot 79^{2} + 45 a\cdot 79^{3} + \left(37 a + 65\right)\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $8$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $6$ | $4$ | $(1,8,3,7)(2,5,6,4)$ | $0$ |
| $8$ | $6$ | $(1,5,6,3,4,2)(7,8)$ | $1$ |
| $6$ | $8$ | $(1,2,8,5,3,6,7,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,8,4,3,2,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.