# Component Groups of J_{0}(N)(**R**) and J_{1}(N)(**R**)

## J_{0}(N)(**R**)

I computed the component groups for N<=500, and here is the data as a list of pairs N, order of component group at infinity.

The first example with nontrivial component group is the elliptic curve J_0(15).

**Order** |
**Number of N<=500 with component group of this order** |

1 |
212 |

2 |
253 |

4 |
0 |

8 |
35 |

>8 |
0 |

This data is intriguing and suggests something interesting is going on. Why doesn't 4 ever occur as the order of a component group? Why nothing bigger than 8 (up to level 500)? Very weird. I bet there's a theorem here waiting to be proved.

## J_{1}(N)(**R**)

I computed the component groups for N<=500, and here is the data as a list of pairs N, order of component group at infinity.

For J_1(N) the data is much harder to compute. I computed it for levels N <= 54

and found the following:

**Order** |
**Number of N<=54 with component group of this order** |

1 |
37 |

2 |
8 |

4 |
4 |

8 |
3 |

16 |
1 |

32 |
1 |

>32 |
0 |

Here component groups of order 4 do occur, and even one as big as 32.