Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(213\)\(\medspace = 3 \cdot 71 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | \(\Q(\sqrt{213}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | even |
| Dirichlet character: | \(\displaystyle\left(\frac{213}{\bullet}\right)\) |
| Projective image: | $C_1$ |
| Projective field: | \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ | \(x^{2} - x - 53\)  . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | \( 5 + 4\cdot 11 + 4\cdot 11^{2} + 2\cdot 11^{3} + 2\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 2 }$ | $=$ | \( 7 + 6\cdot 11 + 6\cdot 11^{2} + 8\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\)
|
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.