Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(2563\)\(\medspace = 11 \cdot 233 \) |
Artin number field: | Galois closure of 8.2.16836267547.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T22 |
Parity: | odd |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.2.2563.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ |
\( 1 + 11\cdot 13^{2} + 13^{3} + 10\cdot 13^{4} + 3\cdot 13^{5} + 10\cdot 13^{6} + 5\cdot 13^{7} +O(13^{8})\)
$r_{ 2 }$ |
$=$ |
\( 11 a + 9 + \left(6 a + 8\right)\cdot 13 + 7\cdot 13^{2} + \left(7 a + 7\right)\cdot 13^{3} + 8 a\cdot 13^{4} + 5\cdot 13^{5} + \left(5 a + 4\right)\cdot 13^{6} + \left(a + 7\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 8 a + 10 + \left(8 a + 8\right)\cdot 13 + 3\cdot 13^{2} + \left(8 a + 9\right)\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} + \left(6 a + 1\right)\cdot 13^{5} + \left(6 a + 1\right)\cdot 13^{6} + \left(6 a + 7\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 7 + \left(6 a + 4\right)\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(5 a + 1\right)\cdot 13^{3} + \left(4 a + 2\right)\cdot 13^{4} + \left(12 a + 10\right)\cdot 13^{5} + \left(7 a + 8\right)\cdot 13^{6} + \left(11 a + 3\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a + 5 + \left(4 a + 9\right)\cdot 13 + \left(12 a + 8\right)\cdot 13^{2} + \left(4 a + 3\right)\cdot 13^{3} + \left(11 a + 6\right)\cdot 13^{4} + \left(6 a + 6\right)\cdot 13^{5} + \left(6 a + 1\right)\cdot 13^{6} + \left(6 a + 7\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a + 1 + \left(11 a + 10\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(10 a + 8\right)\cdot 13^{3} + \left(11 a + 9\right)\cdot 13^{4} + \left(5 a + 11\right)\cdot 13^{5} + \left(7 a + 1\right)\cdot 13^{6} + \left(12 a + 7\right)\cdot 13^{7} +O(13^{8})\)
| $r_{ 7 }$ |
$=$ |
\( 12 + 7\cdot 13 + 3\cdot 13^{2} + 6\cdot 13^{3} + 12\cdot 13^{4} + 6\cdot 13^{5} + 7\cdot 13^{6} + 13^{7} +O(13^{8})\)
| $r_{ 8 }$ |
$=$ |
\( 7 a + 7 + \left(a + 2\right)\cdot 13 + \left(8 a + 11\right)\cdot 13^{2} + 2 a\cdot 13^{3} + \left(a + 11\right)\cdot 13^{4} + \left(7 a + 5\right)\cdot 13^{5} + \left(5 a + 3\right)\cdot 13^{6} + 12\cdot 13^{7} +O(13^{8})\)
| |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ | $-2$ |
$12$ | $2$ | $(1,7)(2,6)(4,8)$ | $0$ | $0$ |
$8$ | $3$ | $(1,2,4)(6,7,8)$ | $-1$ | $-1$ |
$6$ | $4$ | $(1,6,7,4)(2,5,8,3)$ | $0$ | $0$ |
$8$ | $6$ | $(1,5,8,7,3,2)(4,6)$ | $1$ | $1$ |
$6$ | $8$ | $(1,4,3,2,7,6,5,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
$6$ | $8$ | $(1,6,3,8,7,4,5,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |