Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6525\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Artin stem field: | Galois closure of 8.4.47897578125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.145.4t1.b.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-87})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 2x^{6} + 9x^{5} - 45x^{4} + 39x^{3} - 107x^{2} - 258x + 211 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 64\cdot 89 + 68\cdot 89^{2} + 87\cdot 89^{3} + 22\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 9\cdot 89 + 69\cdot 89^{2} + 3\cdot 89^{3} + 31\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 24\cdot 89 + 52\cdot 89^{2} + 69\cdot 89^{3} + 48\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 + 51\cdot 89 + 35\cdot 89^{2} + 48\cdot 89^{3} + 60\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 61 + 59\cdot 89 + 52\cdot 89^{2} + 69\cdot 89^{3} + 77\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 65 + 16\cdot 89 + 7\cdot 89^{2} + 30\cdot 89^{3} + 7\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 75 + 56\cdot 89 + 64\cdot 89^{2} + 6\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 82 + 73\cdot 89 + 5\cdot 89^{2} + 40\cdot 89^{3} + 24\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,6,2,8)(3,5,4,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,2,6)(3,7,4,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,2,8)(3,7,4,5)$ | $0$ |
| $2$ | $8$ | $(1,7,6,3,2,5,8,4)$ | $0$ |
| $2$ | $8$ | $(1,3,8,7,2,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,8,3,2,7,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,6,5,2,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.