Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(1097\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{1097}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | even |
Dirichlet character: | \(\displaystyle\left(\frac{1097}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x - 274 \) . |
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 11\cdot 17 + 16\cdot 17^{2} + 4\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\)
$r_{ 2 }$ |
$=$ |
\( 16 + 5\cdot 17 + 12\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)
| |
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.