# Properties

 Label 1.312.2t1.b.a Dimension $1$ Group $C_2$ Conductor $312$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$312$$$$\medspace = 2^{3} \cdot 3 \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of $$\Q(\sqrt{-78})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: odd Dirichlet character: $$\displaystyle\left(\frac{-312}{\bullet}\right)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{2} + 78$$ x^2 + 78 .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$6 + 9\cdot 19 + 7\cdot 19^{2} + 3\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})$$ 6 + 9*19 + 7*19^2 + 3*19^3 + 16*19^4+O(19^5) $r_{ 2 }$ $=$ $$13 + 9\cdot 19 + 11\cdot 19^{2} + 15\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})$$ 13 + 9*19 + 11*19^2 + 15*19^3 + 2*19^4+O(19^5)

## 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.