Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(256\)\(\medspace = 2^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.16777216.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.4.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{6} + 8x^{4} - 4x^{2} + 1 \) . |
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 72\cdot 73 + 34\cdot 73^{2} + 34\cdot 73^{3} + 49\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 19 + 66\cdot 73 + 37\cdot 73^{2} + 34\cdot 73^{3} + 18\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 23 + 12\cdot 73 + 16\cdot 73^{2} + 36\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 29 + 20\cdot 73 + 49\cdot 73^{2} + 5\cdot 73^{3} + 7\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 44 + 52\cdot 73 + 23\cdot 73^{2} + 67\cdot 73^{3} + 65\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 50 + 60\cdot 73 + 56\cdot 73^{2} + 36\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\) |
$r_{ 7 }$ | $=$ | \( 54 + 6\cdot 73 + 35\cdot 73^{2} + 38\cdot 73^{3} + 54\cdot 73^{4} +O(73^{5})\) |
$r_{ 8 }$ | $=$ | \( 68 + 38\cdot 73^{2} + 38\cdot 73^{3} + 23\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
$2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.