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Questions tagged [elliptic-curves]

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

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Multiplication by an element inducing the identity on a quotient

I am studying the chapter on the associated Grössencharacter of a CM ellipic curve in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (II.9.) and have a question concerning a specific ...
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How to generate a (secure) elliptic curve of composite order?

How to generate a (secure) elliptic curve of order $k$ ($k$ is not be a prime)? Elliptic curves with prime order can be generated via different approaches. I wonder how to generate an elliptic curve ...
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1answer
32 views

Product representation of Weierstrass $\wp$-function

Let $\Lambda = w_1\mathbb Z+w_2\mathbb Z$ be a lattice and $$\wp(z)=\frac1{z^2}+\sum_{w\in\Lambda - 0}\frac1{(z-w)^2}-\frac1{w^2}$$ its associated Weierstrass $\wp$-function. Let $n>1$ be an ...
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1answer
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Solving a Diophantine equation $3a^3+3b^3=a^3+c^3$

A friend of mine gave me the following problem Find all integers solutions to $$3a^3+3b^3=a^3+c^3$$ Of course $(0,0,0)$ is a solution, and I think that there are no others, but I can’t prove it. I ...
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70 views

Find isogeny between two given points

Let $P$ be a point on an elliptic curve $E$ and let $Q = \phi(P)$, where $\phi: E \to E'$ is an isogeny of degree $d$. Given $E, E', P, Q$ and $d$, is it possible to find an isogeny $\phi': E \to E'$,...
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1answer
72 views

Dual of an elliptic curve?

Since an elliptic curve given by the homogenized polynomial $$y^2z=x^3+axz^2+bz^3$$ is a plane projective curve, we can get its dual. From this Wikipedia link, eliminating $p$, $q$, $r$, and $λ$ from ...
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1answer
25 views

Integral Bounds of Part of an Ellipse

I'm looking through my notes, and I've completed this questions integral, getting $ -144 \int cos^4(\theta)$ and due to the fact that $x=4cos(\theta)$ I calculated that the lower bound is $cos^-1(\...
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1answer
24 views

How is scalar multiplication performed on a point in Elliptic Curve Cryptography

can someone please explain how the multiplication(386(0,376)) is performed in the given example in the image. ECC example
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0answers
47 views

Intersection of an elliptic curve and a line over a non-algebraically closed field

I read the definition of the group law on elliptic curves and there's one thing I don't understand. In the link above it is stated that In the projective plane, each line will intersect a cubic at ...
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1answer
49 views

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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1answer
62 views

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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1answer
83 views

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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1answer
46 views

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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Bound on coefficients of Hasse-Weil function

Let $E$ be an elliptic curve over $\mathbb{Q}$. For $n\in \mathbb{N}$, we define $a_n$ as follows. $a_1 = 1$. If $n = p$, $$ a_p = \begin{cases} p+1-|E(\mathbb{F_p})| & E \text{ has good ...
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Frobenius endorphism for elliptic curves

In the Pairing for Beginners book, I read: Frobenius endomorphism $\pi$ for $E$: $\pi : E \rightarrow E, (x, y) \mapsto (x^q,y^q)$ Note: $\pi$ maps any point $E(\overline{\mathbb{F}}_q)$ to $E(\...
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2answers
83 views

Solving $P+P=Q$ on the curve $y^2=x^3-n^2x$

If $P=(x_0,y_0)$ is a rational point on the curve $y^2=x^3-n^2x$, let $Q=2P=P+P=(x_1,y_1)$. Then $x_1$, $x_1+n$ and $x_1-n$ are all rational squares (see for example Ch 1 of the book Elliptic curves ...
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Reduction type at p=2

Let $E:y^2=x^3+Ax+B$ be an integral, minimal model for an elliptic curve over $\mathbb{Q}$. The discriminant of $E$ is $\Delta = -2^4(4A^3+27B^2)$ so $E$ always has bad reduction at $p=2$. A search on ...
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1answer
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Number of integral solutions to elliptic curve $\binom n2=\binom m3$.

I am wondering if there are infinitely many integral solutions to the equation: $$ {n \choose 2} = {m \choose 3}. $$ Also, do the solutions have a general form? From what I know, this is an elliptic ...
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Endomorphism ring of $\mathbb{Z}/n\mathbb{Z}$. (in reference to elliptic curves over a ring)

I know for the group $E(\overline{\mathbb{F}_p})$ where $p$ is prime we have the Frobenius endomorphism, say $\phi$, satisfies the equation $(\phi^2 - [t]\phi + [q])P = [0]P$ in the endomorphism ring ...
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Néron-Ogg-Shafarevich and good ordinary / supersingular reduction

Let $\ell\neq p$ be primes. An elliptic curve $E/\mathbb{Q}_p$ has good reduction if and only if $T_{\ell}(E)$ is unramified. Is there a more refined criterion on $T_{\ell}(E)$ which tells us ...
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Height of a formal group law homomorphism

I am reading about formal group homomorphisms defined over a ring $R$ of characteristic $p > 0$ from Silverman's Arithmetic of Elliptic Curves. Given a homomorphism $f$, he shows that if $f'(0)=0$ ...
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good/bad reduction modulo $p$ in Rational Points on Elliptic Curves vs. Arithmetic of Elliptic Curves

I am trying to understand what an elliptic curve mod $\pi$ vs mod $p$ is. Basically I am confused about the treatment given in Silverman's two books. The definition for mod $p$ in Rational Points of ...
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2answers
38 views

elliptic curve over nonprime finite field $\mathbb{F}_{p^n}$

I am currently trying to conceptualize what an elliptic curve over the finite field $\mathbb{F}_{p^n}$ looks like where $p$ is an odd prime. I have never taken a course on field theory so I am still ...
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1answer
21 views

On this particular elliptic curve, how can I construct a function with a prescribed set of poles and zeros?

Consider the elliptic curve given by $E: Y^2 = X^3-X$ over the field $\overline{\mathbb{F}}_5$. I have computed the $\mathbb{F}_5$-rational points (in projective space, where $(0:1:0)$ is taken as the ...
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For what fields does this morphism from an elliptic curve to the projective line ramify at infinity?

Let $k$ be a field. Consider the curve $X := V_+(X_1^2 X_2 + X_1 X_2^2- X_0^3-X_0^2X_2) \subseteq \mathbb{P}^2_k = \text{Proj}(k[X_0,X_1,X_2])$. Consider the morphism given on functions fields by $k(...
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1answer
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Associativity of formal group law in elliptic curves

In Silverman's AEC there is the following paragraph. Firstly, $F(z_1,z_2)$ lies in $\mathbb{Z}[a_1,...,a_6][[z_1,z_2]]$. Should I treat the $a_i$s as indeterminates or should I treat them lying in ...
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Silverman - MOV attack: why is $E[N] \subset E(\mathbb{F}_{q^d})$ important?

In Silverman's "Arithmetic of Elliptic Curves", the author describes the embedding degree $d$ of the integer $N$ in $\mathbb{F}_q$ as the smallest integer $d$ such that $\mu_N \subset \mathbb{F}_{q^d}^...
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3answers
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Divisor on Elliptic Curve

A divisor on an elliptic curve E is a formal sum of points $$D=\sum_{P\in E}n_P(P)$$ where the $n_P$ are integers only a finite number of which are nonzero. Could anyone please explain what is the ...
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Parallelogram identity to prove that map is linear

Suppose we know $$f(x+y) + f(x-y) = 2f(x) + 2f(y)$$ with $f: G \rightarrow K$ and $G$ an abelian group and $K$ a field. How can we prove that for $$\langle x,y\rangle := \frac{1}{2}(f(x+y)-f(x)-f(y))$...
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1answer
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Polar form of elliptic curve?

My instructor asked us to find the polar form of the elliptic curve defined by the equation $$y^2=x^3+ax+b$$ What I did: Using $x=r\cos\theta$ and $y=r\sin\theta$, I got $$r^2\sin^2\theta=r^3\cos^3\...
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1answer
38 views

What is $l$-th division polynomial of $E$?

I am reading some papers about elliptic curves and I come across the term $l$-th division polynomial of $E$. I don't really know much about field theory and I tried to look for this definition but I'm ...
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Silverman AEC: exercise 3.30

Supposing A is finite abelian with #A = $N^r$ and for every $d | n$ we know #$A[D] = D^r$, with $A[D]$ the subgroup with all elements of order $D$. How do you prove that $A \simeq Z_N^r$ ? I ...
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Finding a curve with maximal number of $\mathbb{F}_{q}$-points [duplicate]

Let $q = p^{n}$ and let $E$ be an elliptic curve. Hasse's bound from the Weil's conjecture tell us that $$ |\sharp E(\mathbb{F}_{q}) - q - 1| \leq 2\sqrt{q} $$ for any $q$. Since this is a uniform ...
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Define the set $E(\mathbb{Q})$ of $\mathbb{Q}$-rational points on an elliptic curve

I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be ...
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$\deg (\alpha) = \#\operatorname{Ker}(\alpha)$ if $\alpha$ is a separable endomorphism of an elliptic curve

In "Elliptic Curves: Number Theory and Cryptography" we see in the proof of proposition 2.21 that for the separable endomorphism $\alpha : E(\overline{K}) \rightarrow E(\overline{K})$ for every point $...
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Automorphisms of stable curves.

I am a bit confused by what Harris and Morrison write about the finiteness condition for stable curves in Moduli of Curves. First the defintions: Definition (2.12) A stable curve is a complete ...
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Elliptic curves over $\mathbb{Q}(\sqrt{2})$ and maps between them

Let $K = \mathbb{Q}(\sqrt{ 2})$. Let $E_1$ be an elliptic curve over $K$. What is the structure of the set of points $E_1(K)$? Assume there is an elliptic curve $E_2$ defined over $K$ and a map $f : ...
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$K$-rational points on an elliptic curve

Let $E$ be an elliptic curve defined over a number field $K$. Denote with $E(K)$ the set of $K$-rational points. Is $E(K)$ always a cyclic group? My attempt: I think this is not true and I am ...
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Show that group $Gal(K/\mathbb{Q})$ acts on $C(K)$ where $C$ is a rational elliptic curve

Let $C$ be a rational elliptic curve, and let $K$ be a Galois extension of $\mathbb{Q}$. a) Prove that for all $P \in C(K)$ and all $\sigma, \tau \in Gal(K/\mathbb{Q})$, $$ \tau(\sigma(P)) = (\...
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1answer
32 views

Why does $\frac{X - aZ}{Z}$ have a double pole at the point $(0 : 1 : 0)$ and not just one (Divisors)?

If I have an Elliptic Curve E and the function $\frac{X - aZ}{Z}$, I would have expected the divisor to be, defining a point $P = (a,b)$ and $-P = (a,-b)$, $div(f) = [P] + [-P] - [\infty]$. Instead ...
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44 views

The n-torsion subgroup $E[n]$ is isomorphic to $Z_n$ x $Z_n$

In Washington's "Elliptic Curves: Number Theory and Cryptography", the proof that $E[n] \simeq Z_n$ x $Z_n$ is concluded by referring to the structure theorem of finite abelian groups ($E[n] \simeq ...
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1answer
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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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1answer
82 views

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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41 views

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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28 views

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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22 views

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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36 views

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 ...