CoCalc Shared Filesmazur-explicit-formula / worksheets / 2016-06-15-084021-ranks-prime-conductor.sagews
Authors: Barry Mazur, William A. Stein
Views : 12

Number of times there are this many curves of prime conductor $p < 2\cdot 10^{13}$, in the range of Benett's data.

1       4617822856
2       54999476
3       6647765
4       386047
5       91229
6       20116
7       5306
8       1697
9       726
10      321
11      116
12      50
13      24
14      12
15      5
16      8
17      3
18      2
19      1
20      1
21      1
v = [[1 ,      4617822856], [2 ,      54999476], [3 ,      6647765], [4 ,      386047], [5 ,      91229], [6 ,      20116], [7 ,      5306], [8 ,      1697], [9 ,      726], [10,      321], [11,      116], [12,      50], [13,      24], [14,      12], [15,      5], [16,      8], [17,      3], [18,      2], [19,      1], [20,      1], [21,      1]]

line2d([[a[0], log(a[1])] for a in v], marker='.', markersize=20, aspect_ratio=1/5, frame=True, ymin=-2, xmin=-1, gridlines=True, figsize=13)

-10/4.

-2.50000000000000
db = CremonaDatabase()

db.number_of_isogeny_classes(37)

2
%time cnts = [db.number_of_isogeny_classes(p) for p in primes(300000)]

CPU time: 1.26 s, Wall time: 2.85 s
set(cnts)

set([0, 1, 2, 3, 4, 5, 7])
w = [[i,cnts.count(i)] for i in [1..7]]
w

[[1, 2745], [2, 275], [3, 98], [4, 24], [5, 5], [6, 0], [7, 1]]
line2d([[a[0], max(-1,log(a[1]))] for a in w], marker='.', markersize=20,  frame=True, gridlines=True)

-6/4.

-1.50000000000000