Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.5120000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.40.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 10x^{6} + 25x^{4} + 20x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 40\cdot 79 + 15\cdot 79^{2} + 69\cdot 79^{3} + 27\cdot 79^{4} + 66\cdot 79^{5} + 40\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 27\cdot 79 + 29\cdot 79^{2} + 8\cdot 79^{3} + 2\cdot 79^{4} + 57\cdot 79^{5} + 64\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 22\cdot 79 + 79^{2} + 61\cdot 79^{3} + 31\cdot 79^{4} + 2\cdot 79^{5} + 40\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 46\cdot 79 + 13\cdot 79^{2} + 46\cdot 79^{3} + 67\cdot 79^{4} + 23\cdot 79^{5} + 61\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 32\cdot 79 + 65\cdot 79^{2} + 32\cdot 79^{3} + 11\cdot 79^{4} + 55\cdot 79^{5} + 17\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 58 + 56\cdot 79 + 77\cdot 79^{2} + 17\cdot 79^{3} + 47\cdot 79^{4} + 76\cdot 79^{5} + 38\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 61 + 51\cdot 79 + 49\cdot 79^{2} + 70\cdot 79^{3} + 76\cdot 79^{4} + 21\cdot 79^{5} + 14\cdot 79^{6} +O(79^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 69 + 38\cdot 79 + 63\cdot 79^{2} + 9\cdot 79^{3} + 51\cdot 79^{4} + 12\cdot 79^{5} + 38\cdot 79^{6} +O(79^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $0$ |
| $2$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.