Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(7168\)\(\medspace = 2^{10} \cdot 7 \) |
| Artin stem field: | Galois closure of 8.4.105226698752.6 |
| 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 $\Q_{ 79 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 42\cdot 79 + 12\cdot 79^{2} + 54\cdot 79^{3} + 27\cdot 79^{4} + 6\cdot 79^{5} + 68\cdot 79^{6} + 67\cdot 79^{7} + 77\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 18\cdot 79 + 48\cdot 79^{2} + 67\cdot 79^{3} + 36\cdot 79^{4} + 24\cdot 79^{5} + 17\cdot 79^{6} + 18\cdot 79^{7} + 64\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 26\cdot 79 + 54\cdot 79^{2} + 52\cdot 79^{3} + 56\cdot 79^{4} + 13\cdot 79^{5} + 63\cdot 79^{6} + 35\cdot 79^{7} + 72\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 63\cdot 79 + 13\cdot 79^{2} + 79^{3} + 31\cdot 79^{4} + 24\cdot 79^{5} + 79^{6} + 62\cdot 79^{7} + 77\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 49 + 15\cdot 79 + 65\cdot 79^{2} + 77\cdot 79^{3} + 47\cdot 79^{4} + 54\cdot 79^{5} + 77\cdot 79^{6} + 16\cdot 79^{7} + 79^{8} +O(79^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 52\cdot 79 + 24\cdot 79^{2} + 26\cdot 79^{3} + 22\cdot 79^{4} + 65\cdot 79^{5} + 15\cdot 79^{6} + 43\cdot 79^{7} + 6\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 71 + 60\cdot 79 + 30\cdot 79^{2} + 11\cdot 79^{3} + 42\cdot 79^{4} + 54\cdot 79^{5} + 61\cdot 79^{6} + 60\cdot 79^{7} + 14\cdot 79^{8} +O(79^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 36\cdot 79 + 66\cdot 79^{2} + 24\cdot 79^{3} + 51\cdot 79^{4} + 72\cdot 79^{5} + 10\cdot 79^{6} + 11\cdot 79^{7} + 79^{8} +O(79^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.