# Properties

 Label 1.751.2t1.1c1 Dimension 1 Group $C_2$ Conductor $751$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $751$ Artin number field: Splitting field of $f= x^{2} - x + 188$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-751}{\bullet}\right)$$

## Galois action

### Roots of defining polynomial

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

### 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.