Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(937024\)\(\medspace = 2^{6} \cdot 11^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.19954863104.2 |
| 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 4.2.21296.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 22x^{4} - 44x^{3} - 44x - 99 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + \left(11 a + 10\right)\cdot 13 + \left(9 a + 7\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(7 a + 10\right)\cdot 13^{4} + \left(5 a + 9\right)\cdot 13^{5} + \left(2 a + 1\right)\cdot 13^{6} + 7 a\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 2 + \left(a + 6\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(2 a + 5\right)\cdot 13^{3} + \left(5 a + 7\right)\cdot 13^{4} + \left(7 a + 7\right)\cdot 13^{5} + \left(10 a + 11\right)\cdot 13^{6} + \left(5 a + 4\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 6 + \left(a + 1\right)\cdot 13 + \left(a + 5\right)\cdot 13^{2} + \left(3 a + 12\right)\cdot 13^{3} + \left(9 a + 1\right)\cdot 13^{4} + 9 a\cdot 13^{5} + \left(10 a + 6\right)\cdot 13^{6} + \left(12 a + 2\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 10 + 4 a\cdot 13 + \left(2 a + 11\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + \left(8 a + 8\right)\cdot 13^{4} + \left(3 a + 7\right)\cdot 13^{5} + \left(2 a + 8\right)\cdot 13^{6} + \left(4 a + 1\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 4\cdot 13 + 5\cdot 13^{2} + 9\cdot 13^{3} + 7\cdot 13^{4} + 11\cdot 13^{5} + 2\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 10 + \left(11 a + 11\right)\cdot 13 + \left(11 a + 4\right)\cdot 13^{2} + \left(9 a + 1\right)\cdot 13^{3} + \left(3 a + 8\right)\cdot 13^{4} + 3 a\cdot 13^{5} + \left(2 a + 7\right)\cdot 13^{6} + 4\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 9 + 5\cdot 13 + 2\cdot 13^{2} + 6\cdot 13^{3} + 11\cdot 13^{4} + 11\cdot 13^{5} + 8\cdot 13^{6} + 6\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a + 4 + \left(8 a + 11\right)\cdot 13 + \left(10 a + 8\right)\cdot 13^{2} + \left(5 a + 8\right)\cdot 13^{3} + \left(4 a + 9\right)\cdot 13^{4} + \left(9 a + 2\right)\cdot 13^{5} + \left(10 a + 7\right)\cdot 13^{6} + \left(8 a + 3\right)\cdot 13^{7} +O(13^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-4$ |
| $12$ | $2$ | $(1,4)(3,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,6,8)(2,4,3)$ | $1$ |
| $6$ | $4$ | $(1,2,4,8)(3,7,6,5)$ | $0$ |
| $8$ | $6$ | $(1,2,6,4,8,3)(5,7)$ | $-1$ |
| $6$ | $8$ | $(1,6,2,5,4,3,8,7)$ | $0$ |
| $6$ | $8$ | $(1,3,2,7,4,6,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.