Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \) |
| Artin number field: | Galois closure of 8.0.70304000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.1098500.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 34\cdot 73 + 23\cdot 73^{2} + 42\cdot 73^{3} + 28\cdot 73^{4} + 45\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 49\cdot 73 + 22\cdot 73^{2} + 38\cdot 73^{3} + 73^{4} + 27\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 64\cdot 73 + 25\cdot 73^{2} + 73^{3} + 70\cdot 73^{4} + 56\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 7\cdot 73 + 65\cdot 73^{2} + 45\cdot 73^{3} + 48\cdot 73^{4} + 58\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 36\cdot 73 + 55\cdot 73^{2} + 47\cdot 73^{3} + 57\cdot 73^{4} + 19\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 + 68\cdot 73 + 3\cdot 73^{2} + 23\cdot 73^{3} + 21\cdot 73^{4} + 22\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 60 + 65\cdot 73 + 5\cdot 73^{2} + 36\cdot 73^{3} + 8\cdot 73^{4} + 35\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 68 + 38\cdot 73 + 16\cdot 73^{2} + 57\cdot 73^{3} + 55\cdot 73^{4} + 26\cdot 73^{5} +O(73^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,2)(4,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,7,2,4)(3,8,6,5)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,7)(3,5,6,8)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,4)(3,5,6,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,2,4)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,4,2,7)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2)(3,5,6,8)(4,7)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,2)(3,8,6,5)(4,7)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,6,2,3)(4,8,7,5)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,6,7,5,2,3,4,8)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,5,4,6,2,8,7,3)$ | $0$ | $0$ |