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

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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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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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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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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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An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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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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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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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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The sum of three colinear rational points is equal to $O$

Show that in an elliptic curve $E/\mathbb{Q}$ the sum of three colinear rational points of it is equal to $O$ exactly when the neutral element of the group $E(\mathbb{Q})$, $O$ is an inflection point ...
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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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127 views

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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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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57 views

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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152 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
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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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141 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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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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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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246 views

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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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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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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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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102 views

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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67 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 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 (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 curve cryptography order

How do I compute an order a a point P on an elliptic curve? My question is specifically in reference to the attached photo. I understand how to do part a but I am totally lost in part b. I don't know ...
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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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108 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 ...
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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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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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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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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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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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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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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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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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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?
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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}) ...
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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 ...
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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 $k$, ...
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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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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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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 ...
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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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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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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 ...
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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 ...