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

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Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
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26 views

$2 \times 2$ matrix representing elliptic curve?

Suppose we have $E/\mathbb{C}$ and we let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda=\mathbb{Z}+\mathbb{Z}\sqrt{5}i$. Suppose also that $\alpha=10+3\sqrt{5}i$. For a basis $\{\...
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Trace and degree of elliptic curve endomorphism?

Let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda = \mathbb{Z} + \mathbb{Z}\sqrt{5}i$. Let $\alpha=10+3\sqrt{5}i$. Show that $\alpha \in$ End$(E)$ and compute the trace and degree of $\...
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36 views

Elliptic curve linear recurrence proof

Let $E/\mathbb{F}_q$ be an elliptic curve with $q=p^m$ for some prime $p$. Let $a_n=q^n+1-\#E(F_{q^n})$ and by convention we let $a_0=2$. Prove that $a_{n+2}=a_1a_{n+1}-qa_n$ for all $n>0$. ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then $...
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25 views

The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
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Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
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36 views

Relation between pullbacks of a degree zero line bundle on an elliptic curve

Let $E$ be an elliptic curve over a field $k$. Let $$\mu:E \times_k E \to E$$ be the addition map on $E$. Furthermore let $p_1,p_2:E \times_k E \to E$ be the two canonical projections and let ${\...
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27 views

Help needed in determining the singularity

Can someone teach me how to determine the singularity of algebraic curve $y^2 =x^3+x^2$. I'll be really grateful and thanks in advance
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help to parametrize $y^2 = x^3 -x^2$

I appreciate if someone could help me to parametrize this equation $y^2 = x^3 -x^2$. Thanks in advance. I used maple to find the solution as $(x,y) = ((t^2-1),(t(t^2-1))$
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$p$ adic modular forms and wide open neighbourhood (e.g. Coleman primitive): is it possible to obtain a holomorphic function?

It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, ...
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Show 2-torsion subgroups are equivalent?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero squarefree element in $k$. Let $E_c/k$ be a curve defined by $cy^2=x^3+ax+b$. Show that the 2-...
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Simple automorphism proof with lattice elliptic curves?

For a lattice $\Lambda_1=\mathbb{Z}+\mathbb{Z}i$, find Aut($E_1$) where $E_1(\mathbb{C})=\mathbb{C}/\Lambda_1$. So I know that End($E$) $ =\{\beta \in \mathbb{C} $| $\beta\Lambda \subseteq \Lambda\...
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Elliptic curve n-torsion point?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, we ...
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53 views

What is more amazing?

$S$ denotes the set of rational points of any curve in the plane. What is more amazing between a) and b)? a) $S$ is dense in the curve $y^2=x^3-2^4\cdot3^3\cdot7^2$ b) $S=\emptyset$ in the curve $...
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40 views

The bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
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22 views

How to show elliptic curve endomorphism is commutative?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-$n$ endomorphism. Show that $[n] \bullet \alpha=\alpha \bullet [n]$, ...
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35 views

Show an elliptic curve is a twist of another curve?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero square free element in $k$. Let ${E_c}/k$ be a curve defined by $cy^2=x^3+ax+b$. Using a linear ...
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23 views

Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
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50 views

Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
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27 views

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
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How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over $\mathbb{Q}$...
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39 views

Show $(x,y) \rightarrow (x,-y)$ is a group homomorphism?

Show that $(x,y) \rightarrow (x,-y)$ is a group homomorphism from $E$ to itself where $E$ is an elliptic curve in Weierstrass form. So $E$ is of the form $y^2=x^3+ax+b$. Would I just show that any ...
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67 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is "...
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27 views

Computing the multiplicative inverse for point addition on an elliptic curve

I'm trying to perform point addition on an elliptic for two points taken from an example in the book "Understanding Cryptography by Christof Paar & Jan Pelzl". The points I'm trying to add are: $$...
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71 views

Quartic Diophantine equation $ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$

About the quartic Diophantine equation: $$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
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Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, $...
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65 views

Isogenous elliptic curves over finite fields have the same number of points

I'm stuck in this question, it is the first part of exercise 5.4 from Silverman - The arithmetic of elliptic curves. Let $C,D$ be two isogenous elliptic curves over a finite field $\mathbb{F}_q$. ...
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Degree $5$ embeddings of genus 1 curves, plucker embedding of a Grassmannian…

Here is something Ravi Vakil says after 19.9F in his notes. He is talking about embeddings of a genus 1 curve $C$ by complete linear series associated to the divisors $O(nP)$. "The beautiful ...
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What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic $L$-...
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Is a projective curve minus finite number of points affine?

The answer to the question in the title is "yes". This is proved for example here, or here (page 5). However, I seem to be able to construct a counterexample. Can you help me find the flaw in my ...
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50 views

question about Galois theory and dimension

I try to understand the proof of the lemma 13.7 of the following article: http://www.cs.nyu.edu/courses/spring05/G22.3220-001/ec-intro1.pdf The lemma says that if $r$ is a rational function which ...
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Points of finite order : Size of the group E/E0

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...
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52 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
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Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
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42 views

ELMO 2012 Shortlist N9

I'll admit that I've made no progress to solve this one. It is way too hard. I guess I must do some stuffs with elliptic curve to solve it but I got nowhere So, here is the problem: Are there ...
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37 views

Hasse theorem proof

I don't understand the following points 1) why is $E$ isomorphic to the $\ker(\phi - 1)$? 2) Why is $\#\ker(\phi - 1) = \deg(\phi - 1)$? The proof is taken from here page 11 https://www.math....
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degree of isogeny understanding

In the context below why is the degree of the Frobenius endomorphism p ?
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$p$-depletion of a modular form

Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$. Let $f$ be a modular form with ...
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29 views

Mordell-Weil group - 2 descent

On my elliptic curve, I have generator P and 2-torsion point T in general. If i compute points nP and nP+T, and substitute these points in my function,z, I noticed that the non-square part are equal(...
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8 divides $\#C(\mathbb{F}_p)$ for $C$ : $y^2=x(x+1)(x-8)$, $p\geq 5$

I was trying to compute the torsion of $C$ : $y^2=x(x+1)(x-8)$ over $\mathbb{Q}$ by using the fact that the order of the torsion divides the order of $C$ reduced modulo any $p\nmid 2\Delta$. For any ...
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Is $H^{1}(k,E[n])$ a subgroup of $H^{1}(k,E)$?

Let $E/k$ be an elliptic curve. Consider $E[n]$ which is a subgroup of $E$. Is it true that $H^{1}(k,E[n])$ is again a subgroup of $H^{1}(k,E)$ in Galois cohomology? I thought that this was true but I'...
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Does adjoining a $p$-power divisor to an elliptic curve, where $p\neq 2$ is a prime, always result in a Galois group of order greater than $2$?

The question says it: Suppose I have a field $K$, whose characteristic is, for simplicity, zero, and an elliptic curve $E$ over $K$, and $x\in E(K)$. Suppose that $p$ is a prime different from $2$, ...
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Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
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Finding quadratic twist of elliptic curve

Given a elliptic curve over $F_p$ with the equation $E : y^2 = x^3 + Ax + B$, I want to find an isomorphous curve (quadratic twist) which can be written in the form $E': y^2 = x^3 + A'x + B'$ where $A'...
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10 views

How to store tables for ECM stage 2

This question about realization ECM stage 2 on GPU. I now that there exists some optimization for the stage 2 of ECM. Namely, let $N$ be a composite number, $q|N$ be a prime, $P=(x_P::z_P)$ be a point ...
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Automorphism of elliptic curve, Vakil 19.10.E

There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's ...
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120 views

On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
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Elliptic curve Schoof algorithm, projective polynomial point coordinates

I'm trying to understand Schoofs algorithm for determining $\#E(F_P)$ of an Elliptic curve $y^2 = x^3 + ax + b$ over $F_P$. For this I'm looking at the implementation of MIRACL: https://github.com/...
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1answer
110 views

Generalizing Fermat's challenge to Frenicle

In 1643, Fermat asked Frenicle et al to find a special Pythagorean triple $a,b,c$ such that for $n=1$, $$a+nb = r_1^2\\ a^2+b^2 = r_2^4\tag1$$ Equivalently, $$\color{blue}{\big((p^2-q^2)^2-(2pq)^2\...