Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(247\)\(\medspace = 13 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of \(\Q(\sqrt{-247}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | odd |
| Dirichlet character: | \(\displaystyle\left(\frac{-247}{\bullet}\right)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{2} - x + 62 \)
|
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 6\cdot 17 + 6\cdot 17^{2} + 12\cdot 17^{3} + 10\cdot 17^{4} +O(17^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 15 + 10\cdot 17 + 10\cdot 17^{2} + 4\cdot 17^{3} + 6\cdot 17^{4} +O(17^{5})\)
| |
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.