Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.4.14580000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 30x^{4} + 45x^{2} + 45 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 25\cdot 61 + 32\cdot 61^{2} + 55\cdot 61^{3} + 26\cdot 61^{4} + 17\cdot 61^{5} + 39\cdot 61^{6} + 50\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 35\cdot 61 + 2\cdot 61^{2} + 26\cdot 61^{3} + 22\cdot 61^{4} + 13\cdot 61^{5} + 2\cdot 61^{6} + 18\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 7\cdot 61 + 36\cdot 61^{2} + 35\cdot 61^{3} + 23\cdot 61^{4} + 44\cdot 61^{5} + 25\cdot 61^{6} + 31\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 3\cdot 61 + 39\cdot 61^{3} + 3\cdot 61^{4} + 41\cdot 61^{5} + 53\cdot 61^{6} + 5\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 57\cdot 61 + 60\cdot 61^{2} + 21\cdot 61^{3} + 57\cdot 61^{4} + 19\cdot 61^{5} + 7\cdot 61^{6} + 55\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 + 53\cdot 61 + 24\cdot 61^{2} + 25\cdot 61^{3} + 37\cdot 61^{4} + 16\cdot 61^{5} + 35\cdot 61^{6} + 29\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 57 + 25\cdot 61 + 58\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} + 47\cdot 61^{5} + 58\cdot 61^{6} + 42\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 60 + 35\cdot 61 + 28\cdot 61^{2} + 5\cdot 61^{3} + 34\cdot 61^{4} + 43\cdot 61^{5} + 21\cdot 61^{6} + 10\cdot 61^{7} +O(61^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
| $2$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $0$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.