Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(227287\)\(\medspace = 167 \cdot 1361 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{-227287}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | odd |
Dirichlet character: | \(\displaystyle\left(\frac{-227287}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x + 56822 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 13 + 17\cdot 31 + 19\cdot 31^{2} + 14\cdot 31^{3} + 21\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 19 + 13\cdot 31 + 11\cdot 31^{2} + 16\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
| |
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.