# Properties

 Label 1.21.2t1.a.a Dimension $1$ Group $C_2$ Conductor $21$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$21$$$$\medspace = 3 \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: $$\Q(\sqrt{21})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: even Dirichlet character: $$\displaystyle\left(\frac{21}{\bullet}\right)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} - x - 5$$  .

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

Roots:
 $r_{ 1 }$ $=$ $$4\cdot 5 + 3\cdot 5^{3} + 4\cdot 5^{4} +O(5^{5})$$ $r_{ 2 }$ $=$ $$1 + 5 + 4\cdot 5^{2} + 5^{3} +O(5^{5})$$

## 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.