Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(1444\)\(\medspace = 2^{2} \cdot 19^{2} \) |
Artin stem field: | Galois closure of 8.2.14301947824.1 |
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 4.2.27436.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 7x^{6} - 7x^{5} + 2x^{4} + 3x^{3} + 4x^{2} - 6x - 2 \) . |
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:
$r_{ 1 }$ | $=$ | \( 2 + 7\cdot 13 + 11\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 6\cdot 13^{5} + 7\cdot 13^{6} + 12\cdot 13^{7} + 7\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})\) |
$r_{ 2 }$ | $=$ | \( 3 a + 7 + \left(4 a + 4\right)\cdot 13 + \left(2 a + 12\right)\cdot 13^{2} + \left(3 a + 1\right)\cdot 13^{3} + \left(6 a + 2\right)\cdot 13^{4} + \left(11 a + 6\right)\cdot 13^{5} + \left(12 a + 7\right)\cdot 13^{6} + \left(12 a + 1\right)\cdot 13^{7} + 5\cdot 13^{8} + \left(12 a + 12\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 3 }$ | $=$ | \( 10 a + 7 + \left(8 a + 8\right)\cdot 13 + 10 a\cdot 13^{2} + \left(9 a + 11\right)\cdot 13^{3} + \left(6 a + 10\right)\cdot 13^{4} + \left(a + 6\right)\cdot 13^{5} + 5\cdot 13^{6} + 11\cdot 13^{7} + \left(12 a + 7\right)\cdot 13^{8} +O(13^{10})\) |
$r_{ 4 }$ | $=$ | \( 7 a + 10 + \left(11 a + 10\right)\cdot 13 + \left(a + 4\right)\cdot 13^{2} + \left(7 a + 10\right)\cdot 13^{3} + \left(3 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(3 a + 12\right)\cdot 13^{6} + \left(7 a + 10\right)\cdot 13^{7} + \left(8 a + 5\right)\cdot 13^{8} + \left(8 a + 6\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 5 }$ | $=$ | \( 10 a + 10 + \left(8 a + 5\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(9 a + 2\right)\cdot 13^{3} + \left(6 a + 5\right)\cdot 13^{4} + \left(a + 11\right)\cdot 13^{5} + 8\cdot 13^{6} + 13^{7} + \left(12 a + 6\right)\cdot 13^{8} + 10\cdot 13^{9} +O(13^{10})\) |
$r_{ 6 }$ | $=$ | \( 12 + 5\cdot 13 + 13^{2} + 6\cdot 13^{4} + 6\cdot 13^{5} + 5\cdot 13^{6} + 5\cdot 13^{8} + 5\cdot 13^{9} +O(13^{10})\) |
$r_{ 7 }$ | $=$ | \( 3 a + 4 + \left(4 a + 7\right)\cdot 13 + \left(2 a + 2\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(6 a + 7\right)\cdot 13^{4} + \left(11 a + 1\right)\cdot 13^{5} + \left(12 a + 4\right)\cdot 13^{6} + \left(12 a + 11\right)\cdot 13^{7} + 6\cdot 13^{8} + \left(12 a + 2\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 8 }$ | $=$ | \( 6 a + 4 + \left(a + 2\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(5 a + 2\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} + \left(11 a + 11\right)\cdot 13^{5} + 9 a\cdot 13^{6} + \left(5 a + 2\right)\cdot 13^{7} + \left(4 a + 7\right)\cdot 13^{8} + \left(4 a + 6\right)\cdot 13^{9} +O(13^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-2$ |
$12$ | $2$ | $(1,6)(2,4)(3,8)$ | $0$ |
$8$ | $3$ | $(1,3,7)(2,5,6)$ | $-1$ |
$6$ | $4$ | $(1,4,6,8)(2,5,3,7)$ | $0$ |
$8$ | $6$ | $(1,5,3,6,7,2)(4,8)$ | $1$ |
$6$ | $8$ | $(1,8,7,2,6,4,5,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
$6$ | $8$ | $(1,4,7,3,6,8,5,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.