Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Artin number field: | Galois closure of 4.0.316368.2 |
| 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 $\Q_{ 17 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 16\cdot 17^{2} + 10\cdot 17^{3} + 3\cdot 17^{4} + 16\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 3\cdot 17 + 17^{2} + 11\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 13\cdot 17 + 15\cdot 17^{2} + 16\cdot 17^{3} + 16\cdot 17^{4} + 5\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 16\cdot 17 + 6\cdot 17^{3} + 13\cdot 17^{4} +O(17^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ | $\zeta_{4}$ |