# Properties

 Label 1.19.3t1.a.a Dimension $1$ Group $C_3$ Conductor $19$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$19$$ Artin number field: Galois closure of 3.3.361.1 Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Dirichlet character: $$\chi_{19}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{3} - x^{2} - 6 x + 7$.

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

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

## Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$

## Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$

The blue line marks the conjugacy class containing complex conjugation.