Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(225\)\(\medspace = 3^{2} \cdot 5^{2} \) |
| Artin number field: | Galois closure of 8.4.56953125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 36\cdot 61 + 7\cdot 61^{2} + 26\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 47\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 3\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 43\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 37 + 17\cdot 61 + 12\cdot 61^{2} + 59\cdot 61^{3} + 42\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 33\cdot 61 + 37\cdot 61^{2} + 20\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 52 + 61 + 60\cdot 61^{2} + 24\cdot 61^{3} + 23\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 56 + 28\cdot 61 + 52\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 58 + 13\cdot 61 + 60\cdot 61^{2} + 27\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(3,5)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,5,8,3)(2,7,4,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,5)(2,6,4,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,3)(2,6,4,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,4,5,6,8,2,3,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,3,4,8,7,5,2)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,2,3,6,8,4,5,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,5,2,8,7,3,4)$ | $0$ | $0$ |