# Properties

 Label 1.1009.3t1.a Dimension 1 Group $C_3$ Conductor $1009$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $1009$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 336 x + 1719$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $6 + 5\cdot 13 + 12\cdot 13^{2} + 13^{3} + 4\cdot 13^{4} + 12\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 2 }$ $=$ $10 + 2\cdot 13 + 3\cdot 13^{2} + 3\cdot 13^{4} + 6\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 3 }$ $=$ $11 + 4\cdot 13 + 10\cdot 13^{2} + 10\cdot 13^{3} + 5\cdot 13^{4} + 7\cdot 13^{5} +O\left(13^{ 6 }\right)$

### 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 values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.