Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(7168\)\(\medspace = 2^{10} \cdot 7 \) |
| Artin stem field: | Galois closure of 8.4.105226698752.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.112.4t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} - 92x^{4} - 112x^{2} + 98 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 86\cdot 113 + 65\cdot 113^{2} + 112\cdot 113^{3} + 30\cdot 113^{4} + 18\cdot 113^{5} + 71\cdot 113^{6} + 12\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 15\cdot 113 + 28\cdot 113^{2} + 98\cdot 113^{3} + 106\cdot 113^{4} + 80\cdot 113^{5} + 60\cdot 113^{6} + 9\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 109\cdot 113 + 81\cdot 113^{2} + 92\cdot 113^{3} + 111\cdot 113^{4} + 34\cdot 113^{5} + 13\cdot 113^{6} + 111\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 46\cdot 113 + 82\cdot 113^{2} + 73\cdot 113^{3} + 34\cdot 113^{4} + 21\cdot 113^{5} + 46\cdot 113^{6} + 82\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 70 + 66\cdot 113 + 30\cdot 113^{2} + 39\cdot 113^{3} + 78\cdot 113^{4} + 91\cdot 113^{5} + 66\cdot 113^{6} + 30\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 3\cdot 113 + 31\cdot 113^{2} + 20\cdot 113^{3} + 113^{4} + 78\cdot 113^{5} + 99\cdot 113^{6} + 113^{7} +O(113^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 81 + 97\cdot 113 + 84\cdot 113^{2} + 14\cdot 113^{3} + 6\cdot 113^{4} + 32\cdot 113^{5} + 52\cdot 113^{6} + 103\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 112 + 26\cdot 113 + 47\cdot 113^{2} + 82\cdot 113^{4} + 94\cdot 113^{5} + 41\cdot 113^{6} + 100\cdot 113^{7} +O(113^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $0$ |
| $2$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.