Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(2312\)\(\medspace = 2^{3} \cdot 17^{2} \) |
| Artin stem field: | Galois closure of 8.4.26261675072.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.136.4t1.b.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 5x^{6} - 25x^{5} + 42x^{4} - 33x^{3} + 3x^{2} + 10x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 11\cdot 89 + 80\cdot 89^{2} + 27\cdot 89^{3} + 69\cdot 89^{4} + 77\cdot 89^{5} + 70\cdot 89^{6} + 88\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 + 30\cdot 89 + 84\cdot 89^{2} + 25\cdot 89^{3} + 47\cdot 89^{4} + 32\cdot 89^{5} + 3\cdot 89^{6} + 35\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 51 + 59\cdot 89 + 65\cdot 89^{2} + 69\cdot 89^{3} + 54\cdot 89^{4} + 68\cdot 89^{5} + 10\cdot 89^{6} + 56\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 32\cdot 89 + 31\cdot 89^{2} + 40\cdot 89^{3} + 38\cdot 89^{4} + 38\cdot 89^{5} + 59\cdot 89^{6} + 62\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 24\cdot 89 + 34\cdot 89^{2} + 49\cdot 89^{3} + 75\cdot 89^{4} + 43\cdot 89^{5} + 58\cdot 89^{6} + 38\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 63 + 28\cdot 89 + 46\cdot 89^{2} + 72\cdot 89^{3} + 51\cdot 89^{4} + 37\cdot 89^{5} + 4\cdot 89^{6} + 61\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 66 + 45\cdot 89 + 2\cdot 89^{2} + 65\cdot 89^{3} + 81\cdot 89^{4} + 28\cdot 89^{5} + 67\cdot 89^{6} + 33\cdot 89^{7} +O(89^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 85 + 33\cdot 89 + 11\cdot 89^{2} + 5\cdot 89^{3} + 26\cdot 89^{4} + 28\cdot 89^{5} + 81\cdot 89^{6} + 68\cdot 89^{7} +O(89^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-2$ |
| $2$ | $2$ | $(3,8)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,7,5,2)(3,6,8,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,5,7)(3,4,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,5,2)(3,4,8,6)$ | $0$ |
| $2$ | $8$ | $(1,8,7,4,5,3,2,6)$ | $0$ |
| $2$ | $8$ | $(1,4,2,8,5,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,8,2,6,5,3,7,4)$ | $0$ |
| $2$ | $8$ | $(1,6,7,8,5,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.