Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(4275\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 19 \) |
| Artin stem field: | Galois closure of 8.8.20560078125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.95.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{57})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 17x^{6} + 16x^{5} + 100x^{4} + x^{3} - 197x^{2} - 152x - 29 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 46\cdot 89 + 60\cdot 89^{2} + 84\cdot 89^{3} + 31\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 74\cdot 89 + 59\cdot 89^{2} + 28\cdot 89^{3} + 67\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 64\cdot 89 + 6\cdot 89^{2} + 8\cdot 89^{3} + 5\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 + 38\cdot 89 + 17\cdot 89^{2} + 21\cdot 89^{3} + 35\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 60 + 79\cdot 89 + 23\cdot 89^{2} + 36\cdot 89^{3} + 68\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 + 74\cdot 89 + 76\cdot 89^{2} + 2\cdot 89^{3} + 7\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 67 + 54\cdot 89 + 16\cdot 89^{2} + 83\cdot 89^{3} + 4\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 71 + 12\cdot 89 + 5\cdot 89^{2} + 2\cdot 89^{3} + 47\cdot 89^{4} +O(89^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,8,7,3)(2,4,6,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,7,8)(2,5,6,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,8,2,7,4,3,6)$ | $0$ |
| $2$ | $8$ | $(1,2,3,5,7,6,8,4)$ | $0$ |
| $2$ | $8$ | $(1,2,8,4,7,6,3,5)$ | $0$ |
| $2$ | $8$ | $(1,4,3,2,7,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.