Graphing

Figure 1 is a graph of the -series for the curve . This curve has conductor 11 which is the curve of smallest conductor.

Figure 1: A graph of the -series for

The circle on the graph is drawn around the point (1,0), which is the critical point with respect to the BSD conjecture. It is critical in the sense that at this point the -series should have the same order of vanishing as the rank of the elliptic curve. On this graph note that the graph does not appear to pass through that point. This would indicate that the rank of the -series at 1 is zero. However, we also know that the rank of this curve is zero. Hence this graph agrees with the BSD conjecture.

Using our data we can compare the -series for various curves of rank 1. This is shown in Figure 2. The curves show in this graph are with conductor 37, with conductor 43, with conductor 53, and with conductor 57.

Figure 2: Curves of rank 1

We can also compare the -series of various curves all of rank 2 (Figure 3).

Figure 3: Curves of rank 2

The conductors of these curves are 389, 433, 1001, and 3185 respectively. Note how the difference in conductors relates to the peak of the -series between and .

We make the similar comparison for some curves of rank 3 in Figure 4. The three curves shown in this Figure are which has conductor 5077, which has conductor 11197, and which has conductor 11642.

Figure 4: Curves of rank 3

Note that as in the case of the curves of rank 2, the graphs are arranged according to conductor. In this case, a curve of higher conductor is always less than a curve of lower conductor. One might be led to turn this observation into a conjecture, and in fact we were at first going to do so. However, closer examination of the evidence shows that while it may sometimes be true, it is not always the case that if the conductor of is larger than that of then , as might be at first believed. One counter example of this can be seen with two curves of rank 2: Take to be and to be . Then both have rank 2 and the conductor of is 563 and the conductor of is 571. However while . Hence the possible conjecture is untrue.

One can also see that the possible conjecture would not hold for curves of rank 4. This can be seen in Figure 5 which is a graph of the curves .

Figure 5: Curves of rank 4

All of the lines in the graph are so close it is hard to make out exactly what is going on in the graph. However, looking directly at the data we can see that the conjecture does not hold. The curve has conductor 501029. At the point the -series has the value 1.558791254529780353650676198. The curve has conductor 545723 which is larger than the first curve. However at the -series for this curve has the value
1.408951738645349791068825739. Hence the conjecture does not hold for .

With the help of some powerful computing power, we can even calculate data for curves of rank 5 or higher. An example of a graph of the -series for a curve of rank 5 is shown in Figure 6. The curve used for this graph is which has rank 5 and conductor 67445803. It was interesting to note that the -series of all of the curves of rank 5 that we graphed looked amazingly similar to the naked eye.

Figure 6: A curve of rank 5

To see what looks like in comparison to we graph both the -series and -series for four curves of different ranks in Figure 7.

Figure 7: Graph of the -series and -series for various curves

In Figure 7, rank()=n. By looking at the graph of the -series we can see that the BSD conjecture is plausible: does not pass through (0,1), seems to pass through with order 1, flattens out so that it could be seen as having order 2, and curve is even flatter, suggesting that it has order 3.