Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(7168\)\(\medspace = 2^{10} \cdot 7 \) |
| Artin number field: | Galois closure of 8.0.105226698752.7 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{7})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 48\cdot 113 + 30\cdot 113^{2} + 22\cdot 113^{3} + 57\cdot 113^{4} + 84\cdot 113^{5} + 45\cdot 113^{6} + 3\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 85\cdot 113 + 62\cdot 113^{2} + 48\cdot 113^{3} + 83\cdot 113^{4} + 8\cdot 113^{5} + 51\cdot 113^{6} + 78\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 14\cdot 113 + 82\cdot 113^{3} + 6\cdot 113^{4} + 17\cdot 113^{5} + 57\cdot 113^{6} + 26\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 103\cdot 113 + 44\cdot 113^{2} + 84\cdot 113^{3} + 41\cdot 113^{4} + 12\cdot 113^{5} + 51\cdot 113^{6} + 75\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 95 + 9\cdot 113 + 68\cdot 113^{2} + 28\cdot 113^{3} + 71\cdot 113^{4} + 100\cdot 113^{5} + 61\cdot 113^{6} + 37\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 96 + 98\cdot 113 + 112\cdot 113^{2} + 30\cdot 113^{3} + 106\cdot 113^{4} + 95\cdot 113^{5} + 55\cdot 113^{6} + 86\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 108 + 27\cdot 113 + 50\cdot 113^{2} + 64\cdot 113^{3} + 29\cdot 113^{4} + 104\cdot 113^{5} + 61\cdot 113^{6} + 34\cdot 113^{7} +O(113^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 109 + 64\cdot 113 + 82\cdot 113^{2} + 90\cdot 113^{3} + 55\cdot 113^{4} + 28\cdot 113^{5} + 67\cdot 113^{6} + 109\cdot 113^{7} +O(113^{8})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,2,3,4,8,7,6,5)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,4,6,2,8,5,3,7)$ | $0$ | $0$ |