File:Distribution of theta p.svg

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日本語: 楕円曲線 E: y^2=x^3+x^2^x の θ_p（p < 500,000） の度数分布。θ_p 素数 p に対して cos θ_p = (p+1-#E(F_p))/2√p で定義される数
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Created in SageMath by the following code: 

E = EllipticCurve('20.a3')

if E.has_cm():
        print("E has Complex Multiplication!")

P = Primes()
prime_max = 500000

theta = []
final_p = 0
for p in P:
        if p >= prime_max:
                break

        if not E.is_good(p):
                continue

        a_p = 1 + p - E.Np(p)
        theta_p = arccos(a_p/2/sqrt(p))
        theta.append(theta_p)

        final_p = p

HH = histogram(theta, bins=41, alpha = 0.7, color = 'skyblue', edgecolor = 'skyblue', density = True, axes_labels=[r'$\theta$', ''])
ymax = HH.get_minmax_data()['ymax']
SS = plot(2/pi*sin(x)^2, (x,0,pi), color = 'purple', legend_label = r'$\dfrac{2}{\pi}\sin^2(x)$')
SS.set_legend_options(shadow=False)
TT = text(r'$E: '+latex(E)+'$\n$p: 2, 3, ..., ' + final_p + '$', (0.1, ymax), horizontal_alignment='left', vertical_alignment = 'top')
YLABEL = text('relative frequency', (-0.3, ymax/3), horizontal_alignment='right', vertical_alignment = 'bottom', rotation=90, fontsize = 'large', color='black')
(HH+SS+TT+YLABEL).save('distribution of theta_p.svg')
Author Hagurekusa

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I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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current00:33, 19 April 2022Thumbnail for version as of 00:33, 19 April 2022567 × 423 (73 KB)Hagurekusa (talk | contribs)Uploaded own work with UploadWizard

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