# Properties

 Label 1.1857.2t1.a.a Dimension $1$ Group $C_2$ Conductor $1857$ Root number $1$ Indicator $1$

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $$1857$$$$\medspace = 3 \cdot 619$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of $$\Q(\sqrt{1857})$$ Galois orbit size: $1$ Smallest permutation container: $C_2$ Parity: even Dirichlet character: $$\displaystyle\left(\frac{1857}{\bullet}\right)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{2} - x - 464$.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $2 + 7 + 5\cdot 7^{2} + 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 5\cdot 7 + 7^{2} + 5\cdot 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right)$

## 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.