# Properties

 Label 1.3_5.2t1.1c1 Dimension 1 Group $C_2$ Conductor $3 \cdot 5$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $15= 3 \cdot 5$ Artin number field: Splitting field of $f= x^{2} - x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-15}{\bullet}\right)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $6 + 6\cdot 17 + 17^{2} + 11\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 10\cdot 17 + 15\cdot 17^{2} + 5\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right)$

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

 Cycle notation $(1,2)$

### Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.