#Data of the minimum discrimiant (from nftables) for degree <= 7
︠0a4c5165-7a6b-45c6-8261-a86c428a5720s︠
min_discs = [[2,0,-3],\
[2,2,5],\
[3,1,-31],\
[3,3,49],\
[4,0,117],\
[4,2,-275],\
[4,4,725],\
[5,1,1609],\
[5,3,-4511],\
[5,5,14641],\
[6,0,-9747],\
[6,2,28037],\
[6,4,-92779],\
[6,6,300125],\
[7,1,-184607],\
[7,3,612233],\
[7,5,-2306599],\
[7,7,20134393]]
len(min_discs)

colors = rainbow(len(min_discs))

i = len(min_discs)-1
P = plot(2,(x,1,min_discs[i][2].abs()*10),color='black',thickness=0.75, title=r'Minkowski bounds for number fields of degree $n$ at most 7 with $r_1$ real places', title_pos=[0.5,1.2])
for i in range(0, len(min_discs)):
    P += plot_loglog((4/pi)^((min_discs[i][0]-min_discs[i][1])/2) * min_discs[i][0].factorial() / min_discs[i][0]^min_discs[i][0] * sqrt(x), (x, min_discs[i][2].abs(), min_discs[i][2].abs()*10), color=colors[i], legend_label=min_discs[i][0:2])

P.show(ymax=28, legend_loc='right', legend_title=r'$[n,r_1]$')

P.save('Minkowski_bounds.pdf', ymax=28, legend_loc='right', legend_title=r'$[n,r_1]$')

P.save('Minkowski_bounds.png', ymax=28, legend_loc='right', legend_title=r'$[n,r_1]$', figsize=[10.24,5.12])

#The above plot shows that one needs only check [n,r_1] with n <=4, r_1 <= 3
︠01e5cc38-c598-48e0-9f02-4b11a7ed8825s︠
import re
R = PolynomialRing(ZZ, 'x')

nr1s = [(n,r1) for n, r1, _ in min_discs if n <= 4 and r1 <= 3]
counts = dict(zip(nr1s, [0]*6))
fields = []
n = 2
r1 = 0
for d in srange(2, 20):
    if not is_fundamental_discriminant(-d):
        continue
    K = QuadraticField(-d)
    if K.minkowski_bound() < 2:
        counts[n,r1] += 1
        fields.append(K)
    else:
        print "Break condition reached at Delta =", -d, "with bound =", K.minkowski_bound().n()
        break
r1 = 2
for d in srange(2, 20):
    if not is_fundamental_discriminant(d):
        continue
    K = QuadraticField(d)
    if K.minkowski_bound() < 2:
        counts[n,r1] += 1
        fields.append(K)
    else:
        print "Break condition reached at Delta =", d, "with bound =", K.minkowski_bound().n()
        break
for n, r1 in nr1s[2:]:
    file = open('T%s%s.gp'%(n,r1),'r')
    for line in file:
        tokens = re.split("[\[\],]", line[1:-1])
        coeffs = tokens[tokens.index('')+1:]
        coeffs = coeffs[:coeffs.index('')]
        coeffs.reverse()
        K = NumberField(R([ZZ(aa) for aa in coeffs]), name='a')
        if K.minkowski_bound() < 2:
            counts[n,r1] += 1
            fields.append(K)
        else:
            print "Break condition reached at Delta = " + tokens[0], "with bound =", K.minkowski_bound().n()
            break

for aa in nr1s:
    print aa, counts[aa]

save(fields,'Number_fields_MK_less_than_2.sobj')