Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Artin stem field: | Galois closure of 8.2.81415168.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.43.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.688.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 8x^{5} + 6x^{3} - 2x^{2} - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 a + 33 + \left(17 a + 1\right)\cdot 37 + \left(10 a + 15\right)\cdot 37^{2} + \left(28 a + 24\right)\cdot 37^{3} + \left(36 a + 8\right)\cdot 37^{4} + \left(6 a + 19\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 22 + \left(24 a + 34\right)\cdot 37 + \left(27 a + 4\right)\cdot 37^{2} + \left(25 a + 36\right)\cdot 37^{3} + \left(28 a + 19\right)\cdot 37^{4} + \left(11 a + 1\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 7 + \left(33 a + 19\right)\cdot 37 + \left(10 a + 31\right)\cdot 37^{2} + \left(14 a + 1\right)\cdot 37^{3} + \left(15 a + 13\right)\cdot 37^{4} + \left(13 a + 2\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 14\cdot 37 + 5\cdot 37^{2} + 15\cdot 37^{3} + 13\cdot 37^{4} + 35\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 + 14\cdot 37 + 30\cdot 37^{2} + 5\cdot 37^{3} + 18\cdot 37^{4} + 18\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 19 + \left(12 a + 32\right)\cdot 37 + \left(9 a + 16\right)\cdot 37^{2} + 11 a\cdot 37^{3} + \left(8 a + 35\right)\cdot 37^{4} + \left(25 a + 19\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 24 a + 22 + \left(3 a + 28\right)\cdot 37 + \left(26 a + 4\right)\cdot 37^{2} + \left(22 a + 11\right)\cdot 37^{3} + \left(21 a + 23\right)\cdot 37^{4} + \left(23 a + 3\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a + 25 + \left(19 a + 1\right)\cdot 37 + \left(26 a + 2\right)\cdot 37^{2} + \left(8 a + 16\right)\cdot 37^{3} + 16\cdot 37^{4} + \left(30 a + 10\right)\cdot 37^{5} +O(37^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $12$ | $2$ | $(1,7)(2,4)(5,6)$ | $0$ |
| $8$ | $3$ | $(1,2,5)(4,7,6)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
| $8$ | $6$ | $(1,6,5,7,2,4)(3,8)$ | $1$ |
| $6$ | $8$ | $(1,4,3,6,7,5,8,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,5,3,2,7,4,8,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.