Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(39\)\(\medspace = 3 \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.2313441.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.39.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 2x^{6} + 3x^{5} - x^{4} + 3x^{3} + 2x^{2} - x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 23\cdot 43 + 32\cdot 43^{2} + 19\cdot 43^{3} + 32\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 21 + 10\cdot 43 + 18\cdot 43^{2} + 6\cdot 43^{3} + 27\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 25 + 7\cdot 43 + 8\cdot 43^{2} + 35\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 27 + 40\cdot 43 + 34\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\) |
$r_{ 5 }$ | $=$ | \( 30 + 3\cdot 43 + 24\cdot 43^{2} + 16\cdot 43^{4} +O(43^{5})\) |
$r_{ 6 }$ | $=$ | \( 31 + 25\cdot 43 + 27\cdot 43^{2} + 26\cdot 43^{3} + 22\cdot 43^{4} +O(43^{5})\) |
$r_{ 7 }$ | $=$ | \( 33 + 16\cdot 43 + 26\cdot 43^{2} + 12\cdot 43^{3} + 40\cdot 43^{4} +O(43^{5})\) |
$r_{ 8 }$ | $=$ | \( 41 + 34\cdot 43^{2} + 36\cdot 43^{3} + 8\cdot 43^{4} +O(43^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
$2$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
$2$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.