Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(177\)\(\medspace = 3 \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of \(\Q(\sqrt{177}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | even |
| Dirichlet character: | \(\displaystyle\left(\frac{177}{\bullet}\right)\) |
| Projective image: | $C_1$ |
| Projective field: | \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ | $ x^{2} - x - 44 $. |
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | $ 2 + 2\cdot 7 + 7^{2} + 3\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 6 + 4\cdot 7 + 5\cdot 7^{2} + 3\cdot 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.