# Properties

 Label 1.13_79.3t1.1c1 Dimension 1 Group $C_3$ Conductor $13 \cdot 79$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $1027= 13 \cdot 79$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 342 x - 2016$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{1027}(971,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $7 + 5\cdot 7^{3} + 4\cdot 7^{4} + 7^{5} +O\left(7^{ 6 }\right)$ $r_{ 2 }$ $=$ $3 + 5\cdot 7 + 5\cdot 7^{4} + 2\cdot 7^{5} +O\left(7^{ 6 }\right)$ $r_{ 3 }$ $=$ $5 + 6\cdot 7^{2} + 7^{3} + 4\cdot 7^{4} + 2\cdot 7^{5} +O\left(7^{ 6 }\right)$

### 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.