Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(473344\)\(\medspace = 2^{8} \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.325660672.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\textrm{GL(2,3)}$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.688.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 6x^{6} - 12x^{5} + 16x^{4} - 10x^{3} - 4x^{2} + 4x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 11 + \left(9 a + 17\right)\cdot 19 + \left(6 a + 12\right)\cdot 19^{2} + \left(14 a + 15\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} + \left(5 a + 17\right)\cdot 19^{5} + \left(2 a + 11\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 7 + \left(9 a + 12\right)\cdot 19 + \left(12 a + 9\right)\cdot 19^{2} + \left(4 a + 4\right)\cdot 19^{3} + 17 a\cdot 19^{4} + \left(13 a + 2\right)\cdot 19^{5} + \left(16 a + 9\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 13\cdot 19 + 5\cdot 19^{2} + 15\cdot 19^{3} + 16\cdot 19^{4} + 17\cdot 19^{5} + 9\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 16 + \left(9 a + 1\right)\cdot 19 + \left(11 a + 11\right)\cdot 19^{2} + \left(9 a + 9\right)\cdot 19^{3} + \left(7 a + 4\right)\cdot 19^{4} + \left(7 a + 9\right)\cdot 19^{5} + \left(3 a + 11\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 + 3\cdot 19 + 12\cdot 19^{2} + 4\cdot 19^{3} + 2\cdot 19^{4} + 19^{5} + 14\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 14 + \left(13 a + 15\right)\cdot 19 + \left(15 a + 4\right)\cdot 19^{2} + \left(4 a + 5\right)\cdot 19^{3} + \left(9 a + 16\right)\cdot 19^{4} + \left(11 a + 17\right)\cdot 19^{5} + \left(14 a + 13\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( a + 15 + \left(9 a + 12\right)\cdot 19 + \left(7 a + 12\right)\cdot 19^{2} + \left(9 a + 7\right)\cdot 19^{3} + \left(11 a + 2\right)\cdot 19^{4} + \left(11 a + 9\right)\cdot 19^{5} + \left(15 a + 7\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a + 7 + \left(5 a + 17\right)\cdot 19 + \left(3 a + 6\right)\cdot 19^{2} + \left(14 a + 13\right)\cdot 19^{3} + \left(9 a + 1\right)\cdot 19^{4} + \left(7 a + 1\right)\cdot 19^{5} + \left(4 a + 17\right)\cdot 19^{6} +O(19^{7})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-4$ |
| $12$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $8$ | $3$ | $(1,4,8)(2,6,7)$ | $1$ |
| $6$ | $4$ | $(1,5,2,3)(4,8,6,7)$ | $0$ |
| $8$ | $6$ | $(1,7,4,2,8,6)(3,5)$ | $-1$ |
| $6$ | $8$ | $(1,8,3,4,2,7,5,6)$ | $0$ |
| $6$ | $8$ | $(1,7,3,6,2,8,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.