Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(1107\)\(\medspace = 3^{3} \cdot 41 \) |
Artin stem field: | Galois closure of 8.2.150730227.6 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T22 |
Parity: | odd |
Determinant: | 1.123.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.1107.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + x^{6} + 5x^{5} - 11x^{4} + 8x^{3} - 7x^{2} + 4x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 15 a + 17 + \left(13 a + 12\right)\cdot 23 + \left(8 a + 21\right)\cdot 23^{2} + \left(21 a + 4\right)\cdot 23^{3} + \left(15 a + 1\right)\cdot 23^{4} + \left(15 a + 4\right)\cdot 23^{5} + \left(5 a + 4\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 2 }$ | $=$ | \( 17 + 8\cdot 23 + 10\cdot 23^{3} + 2\cdot 23^{4} + 5\cdot 23^{5} + 16\cdot 23^{6} +O(23^{7})\) |
$r_{ 3 }$ | $=$ | \( 16 a + 11 + \left(15 a + 20\right)\cdot 23 + 21 a\cdot 23^{2} + \left(14 a + 22\right)\cdot 23^{3} + \left(8 a + 12\right)\cdot 23^{4} + \left(4 a + 3\right)\cdot 23^{5} + \left(3 a + 12\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 4 }$ | $=$ | \( 2 a + 17 + \left(10 a + 13\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(4 a + 13\right)\cdot 23^{3} + 15\cdot 23^{4} + \left(18 a + 19\right)\cdot 23^{5} + \left(17 a + 20\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 5 }$ | $=$ | \( 8 a + 1 + \left(9 a + 2\right)\cdot 23 + \left(14 a + 2\right)\cdot 23^{2} + \left(a + 16\right)\cdot 23^{3} + \left(7 a + 11\right)\cdot 23^{4} + \left(7 a + 19\right)\cdot 23^{5} + \left(17 a + 22\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 6 }$ | $=$ | \( 7 a + 20 + \left(7 a + 12\right)\cdot 23 + \left(a + 5\right)\cdot 23^{2} + \left(8 a + 7\right)\cdot 23^{3} + \left(14 a + 15\right)\cdot 23^{4} + \left(18 a + 3\right)\cdot 23^{5} + \left(19 a + 14\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 7 }$ | $=$ | \( 21 a + 21 + \left(12 a + 8\right)\cdot 23 + \left(2 a + 10\right)\cdot 23^{2} + \left(18 a + 2\right)\cdot 23^{3} + \left(22 a + 11\right)\cdot 23^{4} + \left(4 a + 9\right)\cdot 23^{5} + \left(5 a + 15\right)\cdot 23^{6} +O(23^{7})\) |
$r_{ 8 }$ | $=$ | \( 12 + 12\cdot 23 + 2\cdot 23^{2} + 16\cdot 23^{3} + 21\cdot 23^{4} + 3\cdot 23^{5} + 9\cdot 23^{6} +O(23^{7})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
$12$ | $2$ | $(1,7)(3,4)(5,6)$ | $0$ |
$8$ | $3$ | $(1,6,4)(3,5,7)$ | $-1$ |
$6$ | $4$ | $(1,4,3,7)(2,6,8,5)$ | $0$ |
$8$ | $6$ | $(1,5,4,3,6,7)(2,8)$ | $1$ |
$6$ | $8$ | $(1,2,4,6,3,8,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
$6$ | $8$ | $(1,8,4,5,3,2,7,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.