Vanishing for twists by $d \equiv 0$ modulo $p$

[Rubinstein's web site] has data which looks at the ratio of vanishing with $d = 0$ mod $p$ compared to all d and shows a connection to the group structure of $E(F_p).$ The data is sorted in increasing value of $p$ * (ratio of vanishing). Smaller values tend to go with cyclic groups, larger values with products of two cyclic groups.

The number in the filenames indicates the total number of twists considered in the data set.

For example, here is the header in the $E_{11}$ file

column 1: p * (# vanishing = 0 mod p)/total

column 2: p

column 3: $a_p$

column 4: $N_p = p+1-a_p$

column 5: group structure of $E(F_p)$

and here are some initial entries:

0.6576839438 3 -1 5 : cyclic
0.6594793716 1193 -21 1215 : cyclic
0.6755940597 1301 27 1275 : 255 x 5
0.6793569962 1549 -15 1565 : cyclic
0.7005016233 1637 33 1605 : cyclic
0.7059290235 1409 -15 1425 : cyclic

In the last line here, the 1409 indicates that we are twisting $E_{11}$ by discriminants divisible by 1409. We found that the proportion of these that had rank even positive rank is about 0.7059/1409. The value of $a_{1409}$ for $E_11$ is -15. The value of $N_{1409}$ is 1425. The group of points of $E_{11}$ over the finite field of size 1409 is cyclic.

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