Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(740\)\(\medspace = 2^{2} \cdot 5 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of \(\Q(\sqrt{-185}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| 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 $\Q_{ 3 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 2\cdot 3 + 2\cdot 3^{2} + 3^{4} +O(3^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 2\cdot 3^{3} + 3^{4} +O(3^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |