# Properties

 Label 1.109.3t1.1c2 Dimension 1 Group $C_3$ Conductor $109$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $109$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 36 x + 4$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{109}(45,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $10 + 6\cdot 17 + 4\cdot 17^{2} + 6\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 2\cdot 17^{2} + 5\cdot 17^{3} + 7\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 + 9\cdot 17 + 10\cdot 17^{2} + 5\cdot 17^{3} + 5\cdot 17^{4} +O\left(17^{ 5 }\right)$

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