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.