# Properties

 Label 1.772.2t1.a.a Dimension $1$ Group $C_2$ Conductor $772$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$772$$$$\medspace = 2^{2} \cdot 193$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of $$\Q(\sqrt{-193})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: odd Dirichlet character: $$\displaystyle\left(\frac{-772}{\bullet}\right)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} + 193$$ x^2 + 193 .

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

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