Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(2432\)\(\medspace = 2^{7} \cdot 19 \) |
Artin stem field: | Galois closure of 8.2.28768731136.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T22 |
Parity: | odd |
Determinant: | 1.19.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.4864.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 8x^{6} + 22x^{4} - 16x^{2} - 19 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 12 a + 11 + \left(11 a + 10\right)\cdot 13 + 12 a\cdot 13^{2} + \left(11 a + 6\right)\cdot 13^{3} + \left(2 a + 3\right)\cdot 13^{4} + \left(4 a + 8\right)\cdot 13^{5} + \left(7 a + 12\right)\cdot 13^{6} + \left(10 a + 8\right)\cdot 13^{7} + 4\cdot 13^{8} + \left(3 a + 5\right)\cdot 13^{9} +O(13^{10})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 7\cdot 13 + 11\cdot 13^{2} + 4\cdot 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} + 12\cdot 13^{6} + 13^{7} + 10\cdot 13^{8} + 3\cdot 13^{9} +O(13^{10})\)
| $r_{ 3 }$ |
$=$ |
\( a + 6 + \left(3 a + 5\right)\cdot 13 + \left(12 a + 8\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + \left(3 a + 9\right)\cdot 13^{4} + \left(7 a + 4\right)\cdot 13^{5} + \left(2 a + 2\right)\cdot 13^{6} + \left(3 a + 6\right)\cdot 13^{7} + \left(a + 7\right)\cdot 13^{8} + \left(4 a + 11\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 4 }$ |
$=$ |
\( a + 10 + \left(a + 10\right)\cdot 13 + 13^{2} + \left(a + 5\right)\cdot 13^{3} + \left(10 a + 7\right)\cdot 13^{4} + \left(8 a + 9\right)\cdot 13^{5} + \left(5 a + 2\right)\cdot 13^{6} + \left(2 a + 12\right)\cdot 13^{7} + \left(12 a + 7\right)\cdot 13^{8} + \left(9 a + 7\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 5 }$ |
$=$ |
\( a + 2 + \left(a + 2\right)\cdot 13 + 12\cdot 13^{2} + \left(a + 6\right)\cdot 13^{3} + \left(10 a + 9\right)\cdot 13^{4} + \left(8 a + 4\right)\cdot 13^{5} + 5 a\cdot 13^{6} + \left(2 a + 4\right)\cdot 13^{7} + \left(12 a + 8\right)\cdot 13^{8} + \left(9 a + 7\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 11 + 5\cdot 13 + 13^{2} + 8\cdot 13^{3} + 7\cdot 13^{4} + 9\cdot 13^{5} + 11\cdot 13^{7} + 2\cdot 13^{8} + 9\cdot 13^{9} +O(13^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 12 a + 7 + \left(9 a + 7\right)\cdot 13 + 4\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(9 a + 3\right)\cdot 13^{4} + \left(5 a + 8\right)\cdot 13^{5} + \left(10 a + 10\right)\cdot 13^{6} + \left(9 a + 6\right)\cdot 13^{7} + \left(11 a + 5\right)\cdot 13^{8} + \left(8 a + 1\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 12 a + 3 + \left(11 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(11 a + 7\right)\cdot 13^{3} + \left(2 a + 5\right)\cdot 13^{4} + \left(4 a + 3\right)\cdot 13^{5} + \left(7 a + 10\right)\cdot 13^{6} + 10 a\cdot 13^{7} + 5\cdot 13^{8} + \left(3 a + 5\right)\cdot 13^{9} +O(13^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
$12$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ |
$8$ | $3$ | $(1,7,8)(3,4,5)$ | $-1$ |
$6$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
$8$ | $6$ | $(1,5)(2,4,3,6,8,7)$ | $1$ |
$6$ | $8$ | $(1,8,3,6,5,4,7,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
$6$ | $8$ | $(1,4,3,2,5,8,7,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.