Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.481890304.3 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-7})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} - 3x^{6} + 8x^{5} + x^{4} - 2x^{3} + 23x^{2} - 12x + 2 \) . |
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 11\cdot 67 + 14\cdot 67^{2} + 46\cdot 67^{3} + 67^{4} +O(67^{5})\) |
$r_{ 2 }$ | $=$ | \( 3 + 45\cdot 67 + 12\cdot 67^{2} + 9\cdot 67^{3} + 60\cdot 67^{4} +O(67^{5})\) |
$r_{ 3 }$ | $=$ | \( 10 + 53\cdot 67 + 40\cdot 67^{2} + 52\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\) |
$r_{ 4 }$ | $=$ | \( 25 + 60\cdot 67 + 14\cdot 67^{2} + 25\cdot 67^{3} + 18\cdot 67^{4} +O(67^{5})\) |
$r_{ 5 }$ | $=$ | \( 29 + 58\cdot 67 + 21\cdot 67^{2} + 66\cdot 67^{3} + 67^{4} +O(67^{5})\) |
$r_{ 6 }$ | $=$ | \( 31 + 9\cdot 67 + 64\cdot 67^{2} + 9\cdot 67^{3} + 61\cdot 67^{4} +O(67^{5})\) |
$r_{ 7 }$ | $=$ | \( 50 + 5\cdot 67 + 33\cdot 67^{2} + 32\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\) |
$r_{ 8 }$ | $=$ | \( 53 + 24\cdot 67 + 66\cdot 67^{2} + 25\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
$2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ |
$2$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.