# Properties

 Label 1.7.3t1.a.b Dimension $1$ Group $C_3$ Conductor $7$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$7$$ Artin field: Galois closure of $$\Q(\zeta_{7})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Dirichlet character: $$\chi_{7}(4,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$3 + 2\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})$$ 3 + 2*13 + 5*13^2 + 5*13^3 + 11*13^4+O(13^5) $r_{ 2 }$ $=$ $$5 + 10\cdot 13 + 3\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})$$ 5 + 10*13 + 3*13^2 + 8*13^3 + 12*13^4+O(13^5) $r_{ 3 }$ $=$ $$6 + 4\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})$$ 6 + 4*13^2 + 12*13^3 + 13^4+O(13^5)

## 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 $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.