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

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Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
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Their product is a cubic of a rational number $x$ minus $x$

It is given the integer $6$. Analyze it into two parts such that their product is a cubic of a rational number $x$ minus $x$. $$$$ Let $y$ be the one factor. The other one is $6-y$. We have ...
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Abelian group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3-2$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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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 ...
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50 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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Division on Elliptic Curve points

For dividing two elliptic curve points(mod some value), we use multiplication and modular inverse. Would the reason be due to division being undefined otherwise? The points of an elliptic curve, yield ...
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103 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 ...
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Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
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How Appell-Humbert theorem works in the simplest case of an elliptic curve

Line bundles on complex tori $V/\Lambda$ could be described by a pair $(H, \chi)$, where $H$ is a hermitian form on $V$ s.t. $\operatorname{Im} H(\Lambda, \Lambda) \subset \mathbb{Z}$, and $\chi$ is a ...
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Genus 2 Elliptic curves & their periods

The first part of my question is just a check of my knowledge on elliptic curves. I'm fairly happy with the number theory side of things (torsions, rank, whatever) but is my understanding of the more ...
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1answer
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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). ...
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2answers
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How do I generate group table for elliptic curves over finite fields

Can someone please explain how to generate a group table for an elliptic curve over a finite field? The number of solutions or points are about 16 and it is not possible to do them by adding each ...
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1answer
66 views

Families of Elliptic Curves

I am looking to test some properties of elliptic curves and I would like to have a variety of different families to test. I was wondering if there was, say, a catalogue of the different interesting ...
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35 views

Group tables for elliptic curves over primes

When constructing a group table for an elliptic curve modulo a relatively large prime $p$, say 23, are adding a few points with respect to each other enough to establish symmetry and thereby deduce ...
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180 views

What is an embedding degree of elliptic curve?

I am dealing with MOV algorithm to transform ECDLP to DLP in $GF(p^k)$, but at the first step I have to determine embedding degree k. I have read the definitions of embedding degree, but still I am ...
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Finding Tate-Shafarevich group?

What is the algorithm to find Tate-Shafarevich group of the Mordell's equation $ y^2=x^3-m.$ Thank you in advance.
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Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work ...
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More elliptic curves for $x^4+y^4+z^4 = 1$?

(Note: This has been updated to be similar with this MO post.) There are exactly 22 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{11}$. Noam Elkies showed that $(1)$ as, ...
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Pullback of indecomposable bundles on an elliptic curve

I consider an elliptic curve $\mathcal C$ over $\mathbb{C}$ and the multiplication by $[n]$ map on the curve. Then I consider an indecomposable vector bundle $E$ on $C$. What can I say of the ...
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Fact check: global geometry / topology of moduli space of curves

Question: Is the moduli space of smooth complex curves of genus $g\geq2$ isomorphic to the affine space $\mathbb A_{\mathbb C}^{3g-3}$? (Note: I am not asking about the compactification of this ...
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Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
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1answer
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Parametrization of line bundles over an elliptic curve by points of that curve

Let $E$ be an elliptic curve over an algebraically closed field of characteristic zero, and let $\mathcal{L}$ be a line bundle on $E$ of degree $3$. Suppose, I can present this line bundle as $$ ...
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Point of elliptic curve

How can we calculate the multiple of a point of an elliptic curve? For example having the elliptic curve $y^2=x^3+x^2-25x+39$ over $\mathbb{Q}$ and the point $P=(21, 96)$. To find the point $6P$ ...
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Rational points on an elliptic curve

Consider the following elliptic curve $y^2=(x+1540)(x-508)(x-65024)$. It is trivial that the points $P_1(-1540,0)$, $P_2(508,0)$ and $P_3(65024,0)$ lie on this curve. It is also quite easy to find ...
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1answer
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Was the Wiles's proof of FLT based on elliptic curves or generalized elliptic curves?

I have been told that Wiles's proof of FLT was based on elliptic curves. But yesterday I read from Takeshi Saito's book "Fermat's Last Theorem Basic Tools" that there is so called generalized elliptic ...
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1answer
67 views

Computing number of points in elliptic curve through frobenius endomorphism

I got the following question where I stuck at the moment. Given is the elliptic curve (EC) equation: $E: y^2+3xy+y=x^3+4x+4$ over the finite field ${\bf F}_5$ The first task is now to find out all ...
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Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
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2answers
236 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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1answer
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Basic computation for the degree of an isogeny

I am trying to compute the degree of the isogeny $\phi:E_{1} \to E_{2}$ where $\phi(x,y)=(\frac{y^2}{x^2},\frac{y(b-x^{2})}{x^2})$ with $E_{1} : y^{2} = x^{3} + ax^{2} + bx$, $E_{2} : Y^{2} = X^{3} - ...
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Projective coordinates for point at infinity on elliptic curve

What is the unique characteristic of the projective coordinates of a point at infinity? I am specifically looking for a characteristic on (short) weierstrass curves. I know that the point at infinity ...
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2answers
146 views

Rank of an elliptic curve

How could we compute the rank of an elliptic curve? I looked for a methodoly in my book, but i didn't find anything. Could you give me a hint? I want to find the rank of the curve $Y^2=X^3+p^2X$ ...
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The points are $\mathbb{Z}$-linearly dependent

If $E/\mathbb{Q}$ the elliptic curve $y^2=x^3+x^2-25x+29$ and $$P_1=\left (\frac{61}{4}, \frac{-469}{8}\right ), P_2=\left ( \frac{-335}{81}, \frac{-6868}{729}\right ) , P_3=\left ( 21, 96\right )$$ ...
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1answer
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About fibers of an elliptic fibration.

Consider the following pencil of cubics: $\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of ...
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1answer
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How could we show that the abelian group has $\text{ rank}=0$?

Let $E/\mathbb{Q}$ the elliptic curve $Y^2=X^3+p^2X$ with $p \equiv 5 \pmod 8$. Show that the abelian group $E(\mathbb{Q})$ has $\text{rank}=0$. Could you give me a hint how we could do this? It is ...
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Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
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1answer
159 views

Elliptic curve- Component of point

If $E/ \mathbb{Q}$ elliptic curve in the general Form of Weierstrass and $P=(x,y)$ a rational point of it, show that the first coordinate of the point $2P$ is $$ ...
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A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$

Update (Dec. 22): I have already solved this question with Magma. Recently, I read a paper [1] and saw the following equation: $$u^3−2u^2−2v^3−20v^2+16v=0.$$ The author then got a Weierstrass ...
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Book/lecture notes on algebraic curves

Although there surely is plenty of references on MSE about algebraic curves, my need are very specific and so I will open this topic anyway. I follow this year a course on (hyper)elliptic curves ...
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74 views

Elliptic curves find points with rational coordinates

The elliptic curve $y^2=x^3+3x+4$ has points O,(-1,0) and (0,2). Find five more points with rational coordinates. The answer to this example gives: (0,-2) (5,-12) (5,12) (71/25,744/125) and ...
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Elliptic curves (sum and multiply)

I was wondering if someone could give me some resources on elliptic curve cryptography. Specifically I need to know how to do something like: $y^2=x^3-x+1$ compute $(0,1)⊕(1,1)$ or $y^2=x^3+x^2-x$ ...
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Elliptic curves 2P, 3P

How do I compute 2P, 3P etc? ex: $y^2=x^3+4xmod7$ and I have to compute the order of (2,3)=P and my example says 2P =(0,0) 3P=(2,4) but I don't know how to get these answers?
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Elliptic curves as cubics as discussed in Ravi Vakil's notes

I was reading the section of Ravi Vakil's Algebraic Geometry notes where he discusses elliptic curves. If we let an elliptic curve be $(E,p)$ (Where $p$ is the distinguished point), we have ...
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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 ...
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1answer
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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)? ...
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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 ...
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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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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 = ...
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References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
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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}) ...