Questions tagged [j-invariant]

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Geometric interpretation of j-invariants of supersingular elliptic curves

In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...
146 views

Twisted modular equation

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$ are integral over $\mathbf Z[j]$. Under what conditions is ...
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The degree of the cube root of the $j$-invariant

I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...
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There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$

Given discriminant $d$ and j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the odd prime factors of the ...
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Asymptotic Formula of the coefficients of the q-expansion of the J-Invariant

I'm currently writing my master thesis about the j-invariant and his q-expansion. Now i have the result that the growth of the coefficients is asymptotically c(n) \sim \frac{e^{4\pi \sqrt{n}}}{\...
144 views

From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...
$j$-invariants of elliptic curves over finite fields
Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...
I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then \begin{align*} \frac{j(i)}{1728} &= 1 \\ \frac{-j(\...