Questions tagged [j-invariant]
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21 questions
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The value of the Hauptmodul at CM point
Let $J$ be a classical normalized $j$-invariant (that is, J=j-744).
Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
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Algebraic degrees of $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + \sqrt{-n}}{2}\right)$ and class numbers of $Q(\sqrt{-n})$
Let $n\in\mathbb N$ be squarefree. Denote by $h(n)$ the class number of $Q(\sqrt{-n})$ and by $d_1(n)$ and $d_2(n)$ the degrees of the algebraic numbers $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + ...
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Is it true that all algebraic values of the $j$-invariant have a real Galois conjugate?
This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it ...
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Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality
The $j$-function (a.k.a. $j$-invariant up to a factor $1728$) is most often only considered for arguments where it yields real values.
It is well known that for $a,b,n\in\mathbb N$, $j\left(\dfrac {a+...
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Uniqueness of the $J$ invariant
It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that
$$J(e^{2\pi i/3})...
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What is the value of $j(2\sqrt{-163})$?
My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...
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strange factorization of a simple combination of j-invariant coefficients
I am just curious: is it accidental that $744^2+744\cdot 3-196884\cdot2=162000$ has only the small primes $2$, $3$ and $5$ in its factorization?
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Questions on the $j$-invariant
The j-invariant as a modular function is typically defined
$$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$
since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a ...
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Geometric interpretation of j-invariants of supersingular elliptic curves
In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...
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Twisted modular equation
Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$
are integral over $\mathbf Z[j]$. Under what conditions is ...
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How do modular functions of level $N>1$ transform under the full modular group?
Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...
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Expressing modular functions of level 9 and 32 as rational functions
Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
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Effective bound on the expansion of the $j$-invariant
The $j$-ivariant has the following Fourier expansion
$$j(\tau)=\frac 1q +\sum_{n=0}^{\infty}a_nq^n=\frac{1}{q}+744+196884q+21493760q^2+\cdots.$$
Here is $q=e^{2\pi i \tau}$.
Is there some simple ...
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Zagier's algebraicity of singular moduli
Let $\mathcal M_m\subset M(2,\mathbb Z)$ be the set matrices with determinant $n$. The modular group $\Gamma$ acts on $\mathcal M_m$ from the left and we have the following finite set as a set of ...
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Why the cofficients of the $j$-invariant polynomial are integers and how to compute them?
In the Page 333 of the book “Elliptic curves number theory and cryptogarphy” Lawrence C. Washington.
Given $\tau_i\in H$(up-half plane),and $j(\tau_i)$ is the j-invariant of the lattice $\mathbb Z_\...
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The degree of the cube root of the $j$-invariant
I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...
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There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$
Given discriminant $d$ and j-function $j(\tau)$, I was looking at,
$$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}}}{\...
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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 ...
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$j$-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 ...
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Difference of j-invariant values and the abc conjecture
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(\...
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