Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.1438646272.5 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.56.2t1.b.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.1568.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 8x^{6} - 22x^{4} + 48x^{3} - 44x^{2} + 16x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 21\cdot 71 + 39\cdot 71^{2} + 50\cdot 71^{3} + 36\cdot 71^{4} + 70\cdot 71^{5} +O(71^{6})\)
$r_{ 2 }$ |
$=$ |
\( 6 + 47\cdot 71 + 53\cdot 71^{2} + 70\cdot 71^{3} + 9\cdot 71^{4} + 70\cdot 71^{5} +O(71^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 21 + 62\cdot 71 + 60\cdot 71^{2} + 2\cdot 71^{3} + 48\cdot 71^{4} + 46\cdot 71^{5} +O(71^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 29 + 63\cdot 71 + 3\cdot 71^{2} + 19\cdot 71^{3} + 48\cdot 71^{4} + 6\cdot 71^{5} +O(71^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 34 + 46\cdot 71^{2} + 68\cdot 71^{3} + 64\cdot 71^{4} + 63\cdot 71^{5} +O(71^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 36 + 10\cdot 71 + 45\cdot 71^{2} + 71^{3} + 47\cdot 71^{4} + 65\cdot 71^{5} +O(71^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 43 + 12\cdot 71 + 23\cdot 71^{2} + 24\cdot 71^{3} + 38\cdot 71^{4} + 57\cdot 71^{5} +O(71^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 46 + 66\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 61\cdot 71^{4} + 44\cdot 71^{5} +O(71^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ |
$4$ | $2$ | $(2,6)(3,5)(7,8)$ | $0$ |
$4$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
$2$ | $4$ | $(1,2,4,6)(3,5,8,7)$ | $0$ |
$2$ | $8$ | $(1,7,2,3,4,5,6,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,3,6,7,4,8,2,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.