# Questions tagged [elliptic-curves]

For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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### Is Elliptic Curve Discrete Logarithm Problem NP-Hard or NP-Complete

I have trouble classifying Elliptic Curve Discrete Logarithm Problem as NP-Hard or NP-Complete. Where does ECDLP belong? Any brief comprehensive answer is encouraged. Thanks.
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### What is a supersingular elliptic curve over arbitrary rings

I read in Katz Mazur's book on moduli spaces of elliptic curves that an elliptic curve over an $\mathbb{F}_p$-algebra $R$ is called ordinary if its geometric points are all ordinary. Now the question ...
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### Elliptic curves (Tate normal form?)

I basically have two question, the other question can be found below. Let $E/k$ be an elliptic curve with $P\in E(k)$ a point of order $\geq 4$. Show that $E$ can be described by \begin{equation*} y^...
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### Do we have a criterion for $j$-invariants that gives $CM$ elliptic curves?

It is well-known that every CM elliptic curve has $j$-invariants which are algebraic integers. Is there any criterion to classify all the $j$-invariant that corresponds to $CM$ elliptic curves? For ...
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### Find three points of order two on elliptic curve.

Let $C$ be the cubic curve defined by $y^2z = x^3 -xz^2$ where $O = (0:1:0)$ is an inflection point. Find three points of order two in the group $(C, O, +)$. I know that $2\cdot P = O$ if and only if ...
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### Misprint in Fearless Symmetry by Ash and Gross? Conditions for Elliptic Curves.

On page 104 of the paperback edition of Fearless Symmetry by Avner Ash and Robert Gross, they give one way of thinking of elliptic curves as $y^2=x^3+Ax+B$ wherein $A$ and $B$ can be any fixed ...
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### How to obtain private key of Elliptic curve by factorization?

Okay i am trying to understand how the elliptic curve cryptography can be cracked by using factorization. Am not getting any clues after searching the internet can someone show me how your help will ...
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### Finding all rational points in $x^2+y^2=6$.

I want to find all rational points in the circle $x^2+y^2=6$. This would be easy if I could find one rational point in the circle, however, it's very hard to guess one in this case. However, I don't ...
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### irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication

Currently I am reading the book, Fermat’s Last Theorem written by Darmon, Diamond and Taylor. (You can find this pdf online http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf) ...
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### Bounding rectangle of ellipse

In a stale branch of the code base I'm working on, I found an interesting algorithm to go from the SVG definition of an elliptical arc, to Bézier curves. This is a question about a small but crucial ...
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### Modular Polynomial Arithmetic in Schoof's Algorithm

I've been trying to implement Schoof's Algorithm, and I understand it except for one part. Near the bottom of page 7 of this paper is where my issue is: http://www-users.math.umn.edu/~musiker/schoof....
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### Torsion of elliptic curves and abelian extensions

Let $L/K$ be an abelian $p$-extension of number fields and $E$ be an elliptic curve over $\Bbb Q$. If $E[p](K)=0$, does it follow that $E[p](L)=0$ ? The converse is obviously true, but I don't have ...
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### Sage code to check independence of rational points on elliptic curve

Suppose I have three rational points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ on certain elliptic curve. Then they are linearly independent if and only if the determinant of matrix $(<P_i,P_j>)_{i,j}$ is ...
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### Proof of weak form of Mazur's theorem by Faltings' theorem

Mazur's theorem about elliptic curve claims that there are only finitely many possibilities for the torsion subgroup of a Modell-Weil group of an elliptic curve over $\mathbb{Q}$, and he also gives a ...
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### When do Heegner points lead to a basis points?

For a rank 1 elliptic curve, a rational point can be obtained from Heegner points. When is this rational point a basis point? If sometimes additional work is required to obtain a basis point, is ...
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### How to know the j-invariant of the modular elliptic curve from the modular form?

How do people compute the $j$-invariant of an elliptic curve $E$ over $\mathbb Q$ from the associated modular form $f=\sum_{n=1}^{\infty} a_nq^n$? In other words, how to compute (at least giving some ...
I am looking for an efficient way to generate a random point on an elliptic curve over a finite field, $E(\mathbb{K})$. I know that you can pick a random $x$, compute e.g. in Weierstrass coordinates ...