data-analysis-prime-level.sagews

# Load invertibility mod 2 data. Each line has the form "p r1 r2" where
# p is a prime, r1 is True or False according to whether the matrix of T_2
# acting on S_2(Gamma_0(p), F_2) is invertible, and r2 similarly for T_2+1.

f = open("mod2ranks-500000.txt", "r")
l = []
d = {}
for s in f:
    if s.isspace(): continue
    p, r1, r2 = s.split(" ")
    p = Integer(p); r1 = eval(r1); r2 = eval(r2)
    l.append((p, r1, r2))
    d[p] = (r1, r2)
f.close()
print("Invertibility data loaded successfully.")

Invertibility data loaded successfully.

# Check that each prime appears exactly once.
print([i[0] for i in l] == prime_range(3, 500000))

True

# Test some correlations.
for (p, r1, r2) in l:
    if p%8 == 3 and r1: print("a", p)
    if p%8 != 7 and r2: print("b", p)

('a', 3) ('b', 3) ('b', 5) ('b', 11) ('b', 13) ('b', 19) ('b', 37) ('b', 43) ('b', 67) ('b', 163)

# Compute percentages, prepare plots.

l1 = []; l2 = []; l3 = []; l4 = [];
i1 = 0; j1 = 0; i2 = 0; j2 = 0; i3 = 0; j3 = 0; i4 = 0; j4 = 0
for (p, r1, r2) in l:
    if p%8 == 1:
        i1 += 1
        if not r1: j1 += 1
        if p > 1000:
            l1.append((log(p), j1/i1))
    if p%8 == 5:
        i2 += 1
        if not r1: j2 += 1
        if p > 1000:
            l2.append((log(p), j2/i2))
    if p%8 == 7:
        i3 += 1
        if not r1: j3 += 1
        if p > 1000:
            l3.append((log(p), j3/i3))
        i4 += 1
        if not r2: j4 += 1
        if p > 1000:
            l4.append((log(p), j4/i4))

# Data for p == 1 mod 8, e == 0; p == 5 mod 8, e == 0; p == 7 mod 8, e == 0; p == 7 mod 8, e == 1

print 1.0 * j1/i1, 1.0 * j2/i2, 1.0 * j3/i3, 1.0 * j4/i4
show(points(l1, color='darkgreen', pointsize=2, aspect_ratio=90),
     points(l2, color='darkgreen', pointsize=2, aspect_ratio=50),
     points(l3, color='darkgreen', pointsize=2, aspect_ratio=70),
     points(l4, color='darkgreen', pointsize=2, aspect_ratio=20))

0.167698858138185 0.421660094257959 0.172603003465537 0.479303041971506

# Compute proportion of kernel computations which can be avoided by prescreening using invertibility mod 2.
i=j=0
for (p, r1, r2) in l:
    j += 5
    if r1: i += 3
    if r2: i += 2
print 1.0*i/j

0.387442521125743