# Properties

 Label 1.1113.2t1.a.a Dimension $1$ Group $C_2$ Conductor $1113$ Root number $1$ Indicator $1$

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$1113$$$$\medspace = 3 \cdot 7 \cdot 53$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: $$\Q(\sqrt{1113})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: even Dirichlet character: $$\displaystyle\left(\frac{1113}{\bullet}\right)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} - x - 278$$  .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$3 + 10\cdot 17 + 9\cdot 17^{3} + 11\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$15 + 6\cdot 17 + 16\cdot 17^{2} + 7\cdot 17^{3} + 5\cdot 17^{4} +O(17^{5})$$

## 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.