Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.421654016.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.2744.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} + 7x^{6} - 14x^{5} + 21x^{4} - 21x^{3} + 21x^{2} - 10x + 2 \) . |
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 87\cdot 107 + 23\cdot 107^{2} + 3\cdot 107^{3} + 37\cdot 107^{4} +O(107^{5})\)
$r_{ 2 }$ |
$=$ |
\( 6 + 64\cdot 107 + 11\cdot 107^{2} + 30\cdot 107^{3} + 36\cdot 107^{4} +O(107^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 25 + 72\cdot 107 + 29\cdot 107^{2} + 23\cdot 107^{3} + 26\cdot 107^{4} +O(107^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 61 + 71\cdot 107 + 7\cdot 107^{2} + 14\cdot 107^{3} + 94\cdot 107^{4} +O(107^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 75 + 67\cdot 107 + 43\cdot 107^{2} + 81\cdot 107^{3} + 97\cdot 107^{4} +O(107^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 84 + 4\cdot 107 + 61\cdot 107^{2} + 21\cdot 107^{3} + 47\cdot 107^{4} +O(107^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 88 + 100\cdot 107 + 48\cdot 107^{2} + 59\cdot 107^{3} + 78\cdot 107^{4} +O(107^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 89 + 66\cdot 107 + 94\cdot 107^{2} + 87\cdot 107^{3} + 10\cdot 107^{4} +O(107^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
$4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
$4$ | $2$ | $(1,7)(2,5)(6,8)$ | $0$ |
$2$ | $4$ | $(1,7,6,8)(2,4,5,3)$ | $0$ |
$2$ | $8$ | $(1,5,8,4,6,2,7,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,4,7,5,6,3,8,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.