Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(53\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{53}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | even |
Dirichlet character: | \(\displaystyle\left(\frac{53}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x - 13 \) . |
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 3\cdot 7 + 2\cdot 7^{2} + 4\cdot 7^{3} + 6\cdot 7^{4} +O(7^{5})\) |
$r_{ 2 }$ | $=$ | \( 5 + 3\cdot 7 + 4\cdot 7^{2} + 2\cdot 7^{3} +O(7^{5})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $1$ | ✓ |
$1$ | $2$ | $(1,2)$ | $-1$ |