# Properties

 Label 1.10063.2t1.a.a Dimension 1 Group $C_2$ Conductor $29 \cdot 347$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $10063= 29 \cdot 347$ Artin number field: Splitting field of $$\Q(\sqrt{-10063})$$ defined by $f= x^{2} - x + 2516$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-10063}{\bullet}\right)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 8\cdot 13 + 7\cdot 13^{2} + 7\cdot 13^{3} + 7\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 + 4\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 5\cdot 13^{4} +O\left(13^{ 5 }\right)$

### 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.