# Elliptic curve $Y^2=X^3+X$

Let $$p$$ be prime s.t. $$p=a^2+4b^2, a\equiv 1\,mod\,4$$.

Then, for $$f=X^3+X$$, $$a_E(p)=2a$$?

(Here, $$a_E(p):=p+1-\#\{(x,y)\in\{0,...p-1\}^2|x^3+x\equiv y^2\,mod\,p\}$$)

• What is the context ? The number of points on $y^2=x^3+x\bmod p$ is studied in the theory of elliptic curves with complex multiplication (here by $\Bbb{Z}[i]$) – reuns Jun 11 at 20:33
• Consult OEIS sequence A138515. – Somos Jun 11 at 21:25
• @reuns - this is an exercise in lecture note for undergraduate number theory course (without hint, not hw) and was skipped in class, so the professor probably tried not to use things like Z[i]. I do not know the original reference.. – C.Park Jun 12 at 6:12
• @Somos - I see the sequence, but I don't see how they got that sequence. Is there any way to obtain that? – C.Park Jun 12 at 6:13