# Questions tagged [elliptic-curves]

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

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### Elliptic Curve over a Finite Field, Adding Graphically

I use Mathematica to add two points graphically on the elliptic curve $y^2 = x^3 + 3x + 8$ over $\mathbb{F}_{13}$. Specifically, I'd like to illustrate $(1,8)+(2,10)=(1,5)=(1,-8)$, but on first glance,...
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### Elliptic Curve Division Points

There is a statement about the number of division points, which I've read in a few papers, but it never seems to have any references where it comes from or why it is true. The statement is the ...
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### Relationship between tangents and double roots

I am dealing with the proof of the following Theorem, taken from Dale Husemöller's book Elliptic Curves: I have trouble to understand the following underlined section of the proof: Could you please ...
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### Find a formula for $m$ in terms of $P_m=(x,y)$ and $s$.

We are given this encoding method for elliptic curves where we let $p$ be a prime and $M,s$ positive integers such that $p>Ms$. We let $E$ be the elliptic curve given by $Y^2=f(X)$, where $f(X)$ is ...
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### trace of Frobenius

how can I calculate trace of Frobenius for a single point on an elliptic curve $E(F_{q^{12}})$? I've tried to sum up 12 points that were different powers Frobenius maps but none of the points don't ...
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### Three points on a line after homomorphism (elementary algebra)

While going through the proof of Mordell's theorem on elliptic curves, I came across a certain homomorphism, and the problem is showing that this is indeed a homomorphism. I assure anyone reading this ...
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### Vertical lines in the projective plane $P^2$

How do you get that two vertical lines in $P^2$ intersect at $(0 : 1 : 0)$ or how do you calculate it? If we look at two parallel lines, their point of intersection is at $(1 : s : 0)$ with s as the ...
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### Equivalent definitions of elliptic curves over a scheme

I would like to compare the two definitions of elliptic curves over an arbitrary scheme. Scholze: A morphism $p: E \to S$ of schemes with a section $e: S \to E$ such that $p$ is proper, flat, and all ...
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### Why is the kernel of reduction contained in the image of this isogeny? (From a paper of Cassels)

$\newcommand{\fp}{\mathfrak{p}} \renewcommand{\phi}{\varphi}$ I'm currently trying to understand a paper of Cassels [see page 189 for the relevant content, esp. (4.9)] and I've hit a little snag. The ...
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### Find the Order of an Elliptic Curve

I have an Elliptic Curve represented by the following equation and values: Elliptic Curve: y^2 = x^3 + A*x + B mod M ...
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### Anomalous EC and MOV attack

I'm reading Washington's book about elliptic curves and I am particularly interested about anomalous curves (p. 159): Why should ord($E(Fq)) = q$ prevent the MOV-attack or what is the idea behind ...
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### How important is Weierstrass normal form to the study of elliptic curves?

I'm really not a fan of Weierstrass normal form. Yes, I know that the points of any cubic curve over any field are in bijection with the points on a Weierstrass curve except for a finite number of ...
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### Isomorphism between $\mathbb{P}^{1}$ and $V : X^{2} + Y^{2} = pZ^{2}$

My question is about an exercise from "arithmetic of elliptic curves": Let $$V : X^{2} + Y^{2} = pZ^{2}$$ be a projective vareity in $\mathbb{P}^2$ and $p$ be a prime number. prove that $V$ is ...
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### Number of Rational Points on $C : ax^2 + bxy + cy^2 = dz^2$ over finite field
Let $p \neq 2$ be a prime, let $a,b,c,d \in \mathbb{F}_p$ satisfy $acd \neq 0$, and let $C$ be the conic given by the homogeneous equation $$C : ax^2 + bxy + cy^2 = dz^2.$$ a) If $b^2 \neq 4ac$...