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 8.0.70304000.4 |
| 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 $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 78\cdot 137 + 12\cdot 137^{2} + 111\cdot 137^{3} + 82\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 26\cdot 137 + 115\cdot 137^{2} + 123\cdot 137^{3} + 8\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 53 + 54\cdot 137 + 66\cdot 137^{2} + 107\cdot 137^{3} + 98\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 46\cdot 137 + 93\cdot 137^{2} + 60\cdot 137^{3} + 30\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 83 + 100\cdot 137 + 24\cdot 137^{2} + 2\cdot 137^{3} + 87\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 87 + 88\cdot 137 + 11\cdot 137^{2} + 4\cdot 137^{3} + 135\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 104 + 98\cdot 137 + 48\cdot 137^{2} + 54\cdot 137^{3} + 105\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 134 + 54\cdot 137 + 38\cdot 137^{2} + 84\cdot 137^{3} + 136\cdot 137^{4} +O(137^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(5,6)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,5,8,6)(2,7,4,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,5)(2,3,4,7)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,6)(2,3,4,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,6)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,5)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,3,4,7)(5,6)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7,4,3)(5,6)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,4,8,2)(3,6,7,5)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,4,5,3,8,2,6,7)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,3,6,4,8,7,5,2)$ | $0$ | $0$ |