Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(221\)\(\medspace = 13 \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{221}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | even |
Dirichlet character: | \(\displaystyle\left(\frac{221}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x - 55 \) . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 4\cdot 5 + 3\cdot 5^{2} + 5^{3} + 2\cdot 5^{4} +O(5^{5})\) |
$r_{ 2 }$ | $=$ | \( 1 + 5 + 5^{2} + 3\cdot 5^{3} + 2\cdot 5^{4} +O(5^{5})\) |
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.