Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(100133\)\(\medspace = 11 \cdot 9103 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of \(\Q(\sqrt{100133}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | even |
Dirichlet character: | \(\displaystyle\left(\frac{100133}{\bullet}\right)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{2} - x - 25033 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 16\cdot 29 + 18\cdot 29^{2} + 16\cdot 29^{3} + 8\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 27 + 12\cdot 29 + 10\cdot 29^{2} + 12\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})\) |
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.