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

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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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19 views

degree of isogeny understanding

In the context below why is the degree of the Frobenius endomorphism p ?
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1answer
47 views

$p$-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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24 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 ...
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2answers
55 views

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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208 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
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2answers
40 views

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 ...
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67 views

Are there any other integer points on the elliptic curve $Y^2 = X^3 + 1$ beyond $(-1, 0), (0, \pm 1), (2, \pm 3)$?

The charm of elliptic curves is that given one or two integer points, one can find others by the group law. However the easy to guess points from the title just pump me around trough a cyclic group of ...
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31 views

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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144 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
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77 views

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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45 views

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 ...
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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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25 views

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: ...
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107 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, ...
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183 views

For which integers $a,b$ does $ab-1$ divide $a^3+1$?

A problem I wasn't able to solve: For which values of $a,b\in\mathbb{Z}$ does $ab-1$ divide $a^3+1$? I am looking for every possible solution. Some of them are trivial, like $a=0,b=0$ or ...
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1answer
39 views

Trace of a function on an elliptic curve

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $E$ be an elliptic curve defined over $F$ and $f \in K(E)^{\times}$ be a function. Define the trace of $f$ to be ...
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56 views

Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$. I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman. They presented a ...
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97 views

Special Euler bricks and $x^2(y^2-1)^2+y^2(x^2-1)^2=z^2$

Define, $$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$ Let, $$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$ and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then, ...
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1answer
42 views

Genus of $3x^{3}+4y^{3}+5z^{3}$

I am supposed to calculate the genus of the projective curve $3x^{3}+4y^{3}+5z^{3}$, then use Weil conjecture. I don't know how to calculate the genus, it seems like plugging in formula just give me ...
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Pictures of curves over finite fields with many points

At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$: $(x + y ...
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Weiertrass equation of an elliptic curve.

We know that every elliptic curve is a non-singular $\textbf{cubic}$ projective curve (curve of genus 1), but we can transform this in the Weiertrass form $$y^2 + a_1xy + a_3y = x^3 + a_2x^2 +a_4x ...
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56 views

Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
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2answers
49 views

What happens in elliptic curve primality testing if you cannot find a suitable discriminant?

I'm trying to understand the computational aspect of elliptic curve primality testing (specifically the Atkin-Morain test), and in general, I understand why it works for a prime number. However, when ...
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60 views

How to find the equation of the curve defining the intersection of two quadrics.

Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that ...
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1answer
57 views

Discriminant of Elliptic Curves

In the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = ...
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What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
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35 views

Discriminant of an elliptic curve

I have found different discriminants for general Weierstrass elliptic curves: $y^2 = x^3 + ax + b$ For example, WolframAlpha state it is $-16(4a^3 + 27b^2)$ on their site ...
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25 views

How to find points at infinity in projective coordinates for the $x^2 -3xy-2y^2-x+1=0$

How to find points at infinity in projective coordinates for this algebraic curve $x^2 -3xy-2y^2-x+1=0$ I don't know how to start! Any help will be appreciated.
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How to prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$

Let $f(x) = x^3 +ax+b$ contained in $Q[x]$ prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$ This is what I've so far since $f(x) = (x-A_1)(x-A_2)(x-A_3), A_1,A_2, A_3$ are ...
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1answer
28 views

Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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2answers
443 views

admissible change of variables on elliptic curve

I was wondering whether an admissible change of variables of an elliptic curve given by a Weierstrass equation respects the group law. Let $E$ be defined over a field $K$ given by the equation ...
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45 views

Order of a point on elliptic curve

I am trying to prove that the following elliptic curve has rank=1: $$y^2=x^3+x^2+x+1$$ From the map $$\delta: E(Q)\rightarrow Q(i)^*/(Q(i)^*)^2$$ $$(x,y)\mapsto (x-i) $$ for $(x,y)\neq O$ and ...
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3answers
58 views

Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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60 views

Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
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Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of ...
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1answer
68 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and ...
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32 views

Computing the dual of frobenius endomorphism

There is an exercise in my course Elliptic Curves and I am not sure if I am doing it right. The question is as follows: Let $E$ be the elliptic curve over $\mathbb{F}_2$ given by $Y^2+Y=X^3$. (a) ...
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66 views

Divisor on curve of genus 2

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points ...
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What is the right dual isogeny?

I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of ...
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42 views

Elliptic Curves over Finite Fields as Two Cyclic Groups

Let $E$ be an elliptic curve over $\mathbb{F}_q$. I want to show $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ where $m_1,m_2 \in \mathbb{Z}$ are such that ...
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41 views

Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
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1answer
92 views

Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to, $$x^3+y^3 = Nz^3\tag1$$ for $N=94$ is, $$15642626656646177^3 + ...
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Why are supersingular elliptic curves useful for cryptography?

I don't know very much about cryptography and would like to learn more. I know the basics of RSA alogrithm and how elliptic curves over finite fields can be used to do something similar. But I would ...
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1answer
31 views

How to convert $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$

I'm to figure out how to convert algebraic curve such as $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$. Any help will be appreciated!
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148 views

Characterization of the $m$-torsion points of an elliptic curve.

Let $(E,\mathcal{O})$ be the elliptic curve of equation $$ f=Y^{2}+a_{1}XY+a_{3}Y-X^{3}-a_{2}X^{2}-a_{4}X-a_{6}, $$ $\alpha:K(E)\rightarrow K(E)$ the derivation such that $$ ...
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Primitive of Weierstrass $\wp$

Consider a lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Take the function $\xi(z) = \frac{-1}{z} - \sum_{w \in L\backslash \{0\}} \Big ( \frac{1}{z-w} + \frac{1}{w} + \frac{z}{w^2} \Big )$. Obviously this ...