Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1156\)\(\medspace = 2^{2} \cdot 17^{2} \) |
| Artin stem field: | Galois closure of 8.4.6565418768.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.68.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 5x^{6} + 9x^{5} - 26x^{4} + 35x^{3} - 31x^{2} + 10x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 66\cdot 89 + 77\cdot 89^{2} + 37\cdot 89^{3} + 32\cdot 89^{4} + 73\cdot 89^{5} +O(89^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 + 7\cdot 89 + 88\cdot 89^{2} + 9\cdot 89^{3} + 79\cdot 89^{4} + 23\cdot 89^{5} + 77\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 52 + 2\cdot 89 + 19\cdot 89^{2} + 33\cdot 89^{3} + 59\cdot 89^{4} + 40\cdot 89^{5} + 87\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 33\cdot 89 + 82\cdot 89^{2} + 30\cdot 89^{3} + 33\cdot 89^{4} + 38\cdot 89^{5} + 85\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 67 + 58\cdot 89 + 58\cdot 89^{2} + 79\cdot 89^{3} + 30\cdot 89^{4} + 35\cdot 89^{5} + 65\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 73 + 68\cdot 89 + 87\cdot 89^{2} + 80\cdot 89^{3} + 49\cdot 89^{4} + 37\cdot 89^{5} + 82\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 79 + 59\cdot 89 + 83\cdot 89^{2} + 43\cdot 89^{3} + 47\cdot 89^{4} + 58\cdot 89^{5} + 6\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 81 + 58\cdot 89 + 36\cdot 89^{2} + 39\cdot 89^{3} + 23\cdot 89^{4} + 48\cdot 89^{5} + 39\cdot 89^{6} +O(89^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,6)$ | $0$ |
| $1$ | $4$ | $(1,2,8,6)(3,4,5,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,2)(3,7,5,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,6)(3,7,5,4)$ | $0$ |
| $2$ | $8$ | $(1,4,2,5,8,7,6,3)$ | $0$ |
| $2$ | $8$ | $(1,5,6,4,8,3,2,7)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,8,4,2,3)$ | $0$ |
| $2$ | $8$ | $(1,5,2,7,8,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.