Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Artin number field: | Galois closure of 4.0.78608.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 10\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} + 10\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 2\cdot 13 + 5\cdot 13^{2} + 4\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 10\cdot 13 + 7\cdot 13^{2} + 8\cdot 13^{3} + 4\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 + 2\cdot 13 + 6\cdot 13^{2} + 4\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ | $\zeta_{4}$ |