Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(1255\)\(\medspace = 5 \cdot 251 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{-1255}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | odd |
Dirichlet character: | \(\displaystyle\left(\frac{-1255}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x + 314 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 14 + 4\cdot 31 + 5\cdot 31^{2} + 19\cdot 31^{3} + 10\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 18 + 26\cdot 31 + 25\cdot 31^{2} + 11\cdot 31^{3} + 20\cdot 31^{4} +O(31^{5})\) |
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.