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

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3
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1answer
59 views

Multiplicity of intersection between tangent and elliptic curve

Doubling a point (adding it to itself) on an elliptic curve is done by taking the tangent to the point and calculating the other point where the line intersects the curve. That point is then reflected ...
1
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1answer
30 views

Elliptic curves: Can I replace a coordinate with any modularly equivalent number?

I have a point (x, y) in an elliptic curve group. Suppose y is negative. Can I rewrite it as a positive number if that positive number is equivalent to y (modulo the characteristic of the group)? ...
4
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1answer
58 views

Distribution of points in ellliptic curves over finite fileds

Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in ...
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0answers
26 views

How to compute addition on eliptic curve mod p?

I have a point P=(10, 9) on the curve $y^2 ≡ x^3 + 26$ mod(35) and I am trying to calculate 3P. I know that for when you do P3 = P1+P2 and P1!=P2, you can do $m=\frac{y_2-y_1}{x_2-x_1}$ $x_3 = m^2 ...
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votes
3answers
282 views

What does the Tate module of an elliptic curve tell us?

I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve). I'm having a hard time isolating the ...
1
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1answer
78 views

Let $y^2 = x^3 + Ax + B$ be a curve and $y = m(x - x_1) + y_1$ tangent at $x_1$. Why is $x_1$ then a double root?

Suppose we have a function $y^2 = x^3 + Ax + B$ which we differentiate implicit to find $$\frac {dy} {dx} = \frac {3x^2 + A} {2y}$$ Now suppose we know a point $(x_1,y_1)$ on the curve. Define $$y = ...
3
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1answer
123 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...
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1answer
35 views

Calculating point 2P on an elliptic curve

The equation for the curve is $$y^2=x^3+ax+b$$ and the point in question is $P(x,y)$. We have to verify that the $x$ coordinate of $2P$ is $(x^4-2ax^2-8bx+a^2)/4y^2$. However, the value I get is ...
0
votes
2answers
80 views

Converting equation into Weierstrass form

I have to convert the equation $y^2 +xy +y=x^3 $ by a change of linear variables to the form $Y^2=X^3+aX+b$ where $a$ and $b$ are rational numbers. So far, by completing the square method I've reduced ...
2
votes
1answer
47 views

How to find a solution to the elliptic curve

We know that one solution of the given elliptic curve is (2, 1) and we have to find another rational solution such that $x$ is not equal to 2 by drawing a tangent to the curve at (2, 1). ...
2
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2answers
79 views

Points on elliptic curve over finite field

Find the points on the elliptic curve $y^2 = x^3 + 2x + 2$ in $\mathbb F_{17}$. Do I have to guess a first point and then use an algorithm to spit out all other points?
2
votes
2answers
51 views

Does the Hasse inequality fail for supersingular elliptic curves?

For supersingular elliptic curve $E: y^2+y=x^3 + 2x$ over $\mathbb{F}_{27}$, $\#E\left(\mathbb{F}_{27}\right) = 55$ but $|\#E(\mathbb{F}_{q}) - (q+1)| \leq 2\sqrt{q} \iff 18 \leq \#E(\mathbb{F}_{27}) ...
3
votes
0answers
61 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
4
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0answers
104 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
0
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1answer
37 views

Group law, cubics and Lie group

Let $C$ be a smooth complex cubic in $CP^{2}$. We know that there is a group structure by using the intersection of projective lines (cf. Ried, Undergraduate AG, Section 2), which is really different ...
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0answers
150 views

3D equation of a cone-like shape

Imagine there are two parallel planes (base plane and plane1) in the following image: There is one point on the base plane and there are several points on the plane1. The positions of these points ...
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0answers
38 views

Does this family of complex elliptic curves have a nontrivial section?

Start with the product $\mathbb{C}\times\mathcal{H}$ ($\mathcal{H} = $ upper half plane). Define an action of $\mathbb{Z}^2$ on the left by $(m,n)\cdot(z,\tau) := (z + m\tau + n,\tau)$. Then the ...
0
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1answer
60 views

graduate level introduction to elliptic curve cryptography

I am looking for a good modern book / lecture-notes about elliptic curve cryptography. Does anyone have good recommendations?
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0answers
32 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
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0answers
16 views

Constrained determination of an elliptic curve with marked points

I am trying to determine some equilibrium position of electrostatic charges on Riemann surfaces, and I was wondering if the following problem was a classical type of problem; Let $z_1,...,z_n$ be n ...
3
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0answers
83 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
0
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1answer
20 views

Specific cartesian coordinates of an ellipse

I want to do the following: 1.) Ask user for the vertical and horizontal distances of the ellipse 2.) With this information calculate the circumference 3.) Divide the circumference by the closest ...
2
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1answer
58 views

Clarification of the proof of **Proposition 1.5** in Silverman

I'm studying the proof of Proposition 1.5 on silverman "The Arithmetic of Elliptic Curves" : Proposition: Let $E$ be an elliptic curve. Then the invariant differential $\omega$ associated to a ...
0
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1answer
29 views

Multiply point by scalar in elliptic curve group

I'm trying to understand how to multiply a point by a scalar to get a point in elliptic curve cryptography. Here's an example from my textbook. The group is E257(0, -4). That's shorthand for y2 = x3 ...
3
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1answer
68 views

How many duplication formulas exist for the Mordell curve family $Y^2-X^3=c$?

For the Mordell equation $$ Y^2-X^3 = c, $$ Bachet gave a famous duplication formula which translates one rational solution $(x_1,y_1)$ into a second rational solution $(x_2,y_2)$. Réalis gave a ...
2
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1answer
36 views

How long does it take to double and add points on an elliptic curve?

I have some problems understanding how many multiplications it takes to add or double points on an elliptic curve in Weierstrass form. This link tells that it's 11 and 14, but I don't quite understand ...
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0answers
67 views

Elliptic Curves

I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is ...
3
votes
1answer
68 views

Elliptic curves with twists of zero rank

I am new to field of Elliptic curves. When I was seeing some papers related to this area I have come across elliptic curves having quadratic(some times cubic ) twists with zero rank. What is the ...
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vote
1answer
33 views

How I can calculate the algebraic rank of $C(ℚ)$

Let us consider the elliptic curve $C$ over $ℚ$ in Weierstrass form $$C:y²=x³+1$$ How I can calculate the algebraic rank of $C(ℚ)$.
3
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1answer
59 views

Generators of elliptic curves?

How to find generators to group $E(\mathbb Q) $ of following elliptic curves $E:y^2=x^3-198 $, $E:y^2=x^3-122 $. Thank you in advance.
4
votes
2answers
96 views

Concrete and elementary applications of modular forms to elliptic curves

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery? It's often been asked what elementary ...
2
votes
1answer
60 views

Chapter II Example 3.3 -p.28 Silverman

I have a question about Chapter II Example 3.3 -p.28 in Silverman "Arithmetic of Elliptic Curves". I feel like I'm misreading it and would like clarification. Let $K$ be a field such that ...
4
votes
1answer
121 views

Rational map of a curve to an elliptic curve

If I have a curve given by $$ y^2 = (x^3-1)(x^3-a), $$ how do I find out if there is a rational variable transformation $y=y(s,t)$, $x=x(s,t)$ that maps this curve onto an elliptic curve of the form ...
1
vote
1answer
31 views

Approximating the Rank

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
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0answers
53 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
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0answers
32 views

Ample non-flex on elliptic curve

Say $E$ is a plane cubic, and $p$ is a point on $E$. Riemann-Roch tells us that $\mathcal O_E(3p)$ is very ample. If $p$ is a flex, it's easy to write down the three sections giving an embedding of ...
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0answers
53 views

Klein's invariant and negative discriminant

Let $J = J(\tau)$ be Klein's invariant and let $0 < k < 1$ be the elliptic modulus. It is known that $$J = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2},$$ where ...
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votes
2answers
1k views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
3
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1answer
56 views

The definition of an elliptic curve?

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other ...
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0answers
51 views

Proof of elliptic curves being an abelian group

What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements ...
3
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1answer
60 views

Is the union of all elliptic curves $\mathbb{R}^2$?

An elliptic curve could be written as $$y^2 = x^3 + a x + b \;.$$ Q1. Is it the case that every point $p \in \mathbb{R}^2$ lies on some elliptic curve? Q2. And what is the natural ...
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0answers
70 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
4
votes
1answer
91 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
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1answer
67 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
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0answers
29 views

The correct formula for lambda when point doubling?

When doubling a point on an elliptic curve, we use $\lambda$. But the equation I found in my book(Silverman and Tate, Rational Points on Elliptic Curves) isn't the same as the one I found when looking ...
4
votes
1answer
59 views

First problem in Silverman's Arithmetic of Elliptic Curves

I started working through Silverman's Arithmetic of Elliptic Curves. For some reason it looks like the first problem in the first chapter is the hardest problem in the whole chapter or I'm completely ...
1
vote
1answer
61 views

Why is reduction modulo $p$ a group homomorphism on Elliptic Curves?

I am reading A. Knapp's book on elliptic curves right now. In Proposition 5.6 the author wants to prove that the reduction map (modulo $p$, where $p$ does not divide the discriminant) of an elliptic ...
4
votes
1answer
208 views

Cube of an integer

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers. Prove that $xyz$ is cube of some integer number. I was wondering about giving a parametrization for the rational points on ...
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0answers
45 views

Quick explanation of $\Gamma \tau$ notation?

I hope you can help me by quickly explaining the following notation: $\Gamma \tau$. This notation is encountered in A First Course in Modular Forms by Fred Diamond and Jerry Shurman (love the book by ...
2
votes
1answer
52 views

class number of pure cubic fields and elliptic curves

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will ...