# Properties

 Label 1.2033.2t1.a.a Dimension $1$ Group $C_2$ Conductor $2033$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$2033$$$$\medspace = 19 \cdot 107$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of $$\Q(\sqrt{2033})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: even Dirichlet character: $$\displaystyle\left(\frac{2033}{\bullet}\right)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} - x - 508$$ x^2 - x - 508 .

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$2 + 8\cdot 11 + 5\cdot 11^{2} + 4\cdot 11^{3} + 4\cdot 11^{4} +O(11^{5})$$ 2 + 8*11 + 5*11^2 + 4*11^3 + 4*11^4+O(11^5) $r_{ 2 }$ $=$ $$10 + 2\cdot 11 + 5\cdot 11^{2} + 6\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})$$ 10 + 2*11 + 5*11^2 + 6*11^3 + 6*11^4+O(11^5)

## Generators of the action on the roots $r_{ 1 }, r_{ 2 }$

 Cycle notation $(1,2)$

## 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.