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 8.0.14580000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.20.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 15x^{6} + 60x^{4} + 90x^{2} + 45 \)
|
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 169\cdot 179 + 76\cdot 179^{2} + 130\cdot 179^{3} + 17\cdot 179^{4} + 65\cdot 179^{5} + 108\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 65\cdot 179 + 136\cdot 179^{2} + 29\cdot 179^{3} + 99\cdot 179^{4} + 92\cdot 179^{5} + 35\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 + 104\cdot 179 + 57\cdot 179^{2} + 69\cdot 179^{3} + 88\cdot 179^{4} + 89\cdot 179^{5} + 65\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 78\cdot 179 + 19\cdot 179^{2} + 37\cdot 179^{3} + 159\cdot 179^{4} + 114\cdot 179^{5} + 45\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 93 + 100\cdot 179 + 159\cdot 179^{2} + 141\cdot 179^{3} + 19\cdot 179^{4} + 64\cdot 179^{5} + 133\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 138 + 74\cdot 179 + 121\cdot 179^{2} + 109\cdot 179^{3} + 90\cdot 179^{4} + 89\cdot 179^{5} + 113\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 + 113\cdot 179 + 42\cdot 179^{2} + 149\cdot 179^{3} + 79\cdot 179^{4} + 86\cdot 179^{5} + 143\cdot 179^{6} +O(179^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 163 + 9\cdot 179 + 102\cdot 179^{2} + 48\cdot 179^{3} + 161\cdot 179^{4} + 113\cdot 179^{5} + 70\cdot 179^{6} +O(179^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $2$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.