Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(145\)\(\medspace = 5 \cdot 29 \) |
| Artin number field: | Galois closure of 8.4.15243125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.121945.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 35\cdot 139 + 67\cdot 139^{2} + 29\cdot 139^{3} + 61\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 46 + 33\cdot 139 + 70\cdot 139^{2} + 108\cdot 139^{3} + 130\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 91\cdot 139 + 134\cdot 139^{2} + 48\cdot 139^{3} + 84\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 39\cdot 139 + 67\cdot 139^{2} + 129\cdot 139^{3} + 105\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 11\cdot 139 + 3\cdot 139^{2} + 122\cdot 139^{3} + 53\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 56 + 69\cdot 139 + 27\cdot 139^{2} + 127\cdot 139^{3} + 122\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 62\cdot 139 + 133\cdot 139^{2} + 25\cdot 139^{3} + 12\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 96 + 74\cdot 139 + 52\cdot 139^{2} + 103\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(2,4)(5,6)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,8,7,3)(2,6,4,5)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,7,8)(2,5,4,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(2,5,4,6)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,6,4,5)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,7)(2,6,4,5)(3,8)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,7)(2,5,4,6)(3,8)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8,7,3)(2,5,4,6)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,2,7,4)(3,5,8,6)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,6,3,2,7,5,8,4)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,8,6,7,4,3,5)$ | $0$ | $0$ |