Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.1257728.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.68.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.19652.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{5} - 4x^{4} + 3x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O(149^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O(149^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O(149^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O(149^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O(149^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O(149^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(2,3)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,5,4,6)(2,7,3,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,4,5)(2,8,3,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,7,3,8)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,8,3,7)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,4,6)(2,3)(7,8)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,4,5)(2,3)(7,8)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5,4,6)(2,8,3,7)$ | $0$ |
| $4$ | $4$ | $(1,8,4,7)(2,6,3,5)$ | $0$ |
| $4$ | $8$ | $(1,2,6,8,4,3,5,7)$ | $0$ |
| $4$ | $8$ | $(1,8,5,2,4,7,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.