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

learn more… | top users | synonyms

2
votes
1answer
77 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
2
votes
1answer
50 views

An isogeny of elliptic curves induces a $\mathbb{Z}_l$-linear map

On page 89 of The Arithmetic of Elliptic Curves (second edition), Silverman says: Let $\phi:E_1\rightarrow E_2$ be an isogeny of elliptic curves. Then $\phi$ induces maps $\phi:E_1[l^n]\rightarrow ...
4
votes
1answer
260 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
1
vote
1answer
58 views

Show that there exists a constant $c$ such that for all $n \in \mathbb{Z}_+$ one has $\#\{\omega \in L\,\vert\,n \leq |\omega| \leq n + 1\} \leq cn$.

Let $L \subset \mathbb{C}$ be a lattice (i.e. $L = \{n\omega_1 + m\omega_2 \,\vert\, \omega_1, \omega_2 \in \mathbb{C},\, \omega_1 / \omega_2 \not\in \mathbb{R}, \, n,m \in \mathbb{Z}\}$). Show ...
5
votes
1answer
84 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
3
votes
1answer
56 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
7
votes
2answers
233 views

why say complex multiplication of elliptic curves is beautiful

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. Just as the title asked. ...
2
votes
2answers
77 views

ODE using Weierstrass's P function

I need a hint for the following problem. "Solve $(x')^2=x^3 − 3x^2 − 4x + 12$ with the initial with initial condition $x(0)=3$". I know I should somehow use Weierstrass's $P$ function because it ...
4
votes
2answers
83 views

Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
2
votes
1answer
53 views

Prove that $E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}$.

Let $E$ be an elliptic curve over $\mathbb{C}$. We know that $E(\mathbb{C}) \cong \mathbb{C}/L$ (this is a group isomorphism) for some lattice $L \subset \mathbb{C}$. Using this fact prove that ...
7
votes
2answers
135 views

$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
3
votes
1answer
52 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
1
vote
1answer
34 views

Computing fractions Weierstrass curves and DLP problem

I am preparing for a crypto exam by making an old practise exam. I got stuck on the following assignment. I got this weierstrass curve The curve $y^2 = x^3$ is not an elliptic curve over $F_{71}$ but ...
1
vote
1answer
85 views

Finding all Points on a Edwards curve

I need to find all affine points on the Edwards curve: $x^2 + y^2 = 1 - 5x^2y^2$ over $F_{13}$ I tackle this by transforming the equation to: $y^2 = \frac{1-x^2}{1+5x^2}$ I then go from x = 0 to ...
0
votes
1answer
32 views

Addition on an Elliptic Curve and Modular Arithmetic involving fractions

I'm having a bit of an issue with addition on elliptic curves. For example, I've been given the curve $Y^2 = X^3 + 2X + 1$, working modulo 3. Now, say I want to add the point $(1,2)$ with itself. To ...
2
votes
0answers
60 views

Parametrization of this elliptic curve

What's the simplest way to parametrize the curve given by the equation $$y^2 = (x^2-a^2)^2 - b^2,$$ namely simple functions (polynomials?) $x(z)$, $y(z)$, that would satisfy the above relation. This ...
1
vote
1answer
29 views

Let E be defined over Fq and let n ≥ 1. Show that E(Fq)[n] and E(Fq)/nE(Fq) have the same order.

Let E be an Elliptic Curve defined over $F_q$ and let n ≥ 1. Show that $E(F_q)[n]$ and $E(F_q)$/$nE(F_q)$ have the same order. I feel like this is obvious. The n-th torsion group $E(F_q)[n]$ ...
0
votes
0answers
25 views

Could a hyper elliptic curve of degree 5 admit only three ordinary double points?

The definition for hyper elliptic curves is those curves which admit a ramified double cover of $P^1$. The given degree five is for some homogeneous equations of degree five that defines the curve, ...
2
votes
2answers
90 views

Reason behind standard names of coefficients in long Weierstrass equation

A long Weierstrass equation is an equation of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, ...
3
votes
3answers
102 views

The group structure of elliptic curve over $\mathbb F_p$

I want to find the group of the elliptic curve $y^2=x^3-x$ over $\mathbb F_p$ for all primes $p \le 29$. But I know only 1 fact about the structure of this group: $E(\mathbb F_p)=\mathbb Z/m \mathbb Z ...
0
votes
1answer
31 views

Explanation and validation of point adding/doubling on elliptic curves

I'd like to implement point multiplication on elliptic curves over prime fields. My problem is that I've found different definition how to do it. At adding: the second parameter of the result is not ...
2
votes
1answer
238 views

Trace of Frobenius of elliptic curve is integer

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...
4
votes
1answer
57 views

Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
1
vote
2answers
102 views

How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
4
votes
3answers
231 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
1
vote
2answers
65 views

What does the $[(0 : 3 : 1)]$ means in Sage.

I tried to solve the integer points of $y(y+2)=x^3+(x+3)(x+5)$ by using Sage's command E.integral_points(). Its output was $[(0 : 3 : 1)]$. I tried that ...
0
votes
1answer
67 views

Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2

Problem: show that an elliptic curve over a field of char 2 has nontrivial 3-torsion points Method: I used SAGE to unwind the duplication formula for an elliptic curve given in short Weierstrass form ...
9
votes
4answers
163 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
3
votes
1answer
51 views

Mordell-Weil rank bound

Given an elliptic curve $y^2 = x(x^2 + bx + c)$ is a non-singular curve, say $c > 0$ and $b^2 - 4c > 0$. Can we show the bound on the rank $r$ in terms of $\nu(c)$ and $\nu(b^2 - 4c)$ without ...
0
votes
0answers
37 views

Constant in an inequality for a height of an elliptic curve

I am trying to find explicitly a constant $\kappa$ in an inequality for the height of an elliptic curve. Suppose the curve $E$ is defined by $y^2 = x^3 - kx$ with $k \neq 0$, the curve is defined over ...
4
votes
1answer
125 views

Why are mathematicians more interested in elliptic curves than other algebraic curves?

Why are mathematicians more interested in elliptic curves than other algebraic curves? There must be some reason that motivates mathematicians to research elliptic curves specifically.
0
votes
0answers
30 views

What is some prerequisites to start reading Knapp's elliptic curve book?

I have a pdf of Knapp's elliptic curve, it looks a lot more elemantary to other books on the same subject. However, what is some prerequisites to start reading Knapp's elliptic curve book?
1
vote
1answer
39 views

Groups of rational points invariant birational transformation

I was reading Silverman and Tate's Rational Points on Elliptical Curves, and it said something along the lines of Birational transformation preserves the structure of the groups of the points ...
0
votes
1answer
84 views

How is this an isomorphism?

$\newcommand\O{\mathcal O}$I was reading Silverman and Tate's Rational Points on Elliptical Curves. In page 21 of the same book it was written We also want to mention that there is nothing ...
2
votes
1answer
51 views

Coefficients of an elliptic curve for which the torsion group is trivial

Consider an elliptic curve in the short Weierstrass form $$ y^2 = x^3 + bx + c, $$ defined over rational numbers ($b,c$ are integers). My goal is to provide an example of congruence relations on $b$ ...
1
vote
0answers
98 views

Transform an elliptic-curve into different models

I am a university student and in this semester I have a course in elliptic curves. There is an exercise for which I cannot give a solution, however given the difficulty of the course I think it is not ...
2
votes
1answer
37 views

Height function constants

Consider an elliptic curve defined over the field of rational numbers and given by $$\mathcal{E}_n: y^2=x^3-kx,\ k \ne 0$$ Let $B = \left(\dfrac{r}{s^2},\dfrac{t}{s^3}\right)$ with $r,s,t$ coprime. ...
0
votes
1answer
101 views

Elliptic Curve: Deduce the formula for doubling a point

Given an elliptic curve $E=\{ (x,y) \in \mathbb{F}_q^2 | y^2=x^3+ax+b \}$. Now deduce the general equation for doubling a point $P:=(x,y) \in E$. --- Firstly I constructed the function f ...
2
votes
2answers
286 views

order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$

It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$. When I tried to get the elements of $E = \{(x,y) \in ...
8
votes
2answers
85 views

How to show there exists no solution to a discrete logarithm problem on an Elliptic Curve?

The exact problem is to show that $\nexists$k such that $k(1,2) = (4,5)$ on the elliptic curve defined by $\widetilde{E}: y^2 = x^3 -14x + 17$ over $\mathbb Q$. Background: E: $y^2 = x^3 + 3$ over ...
2
votes
2answers
122 views

Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ ...
4
votes
1answer
78 views

Automorphisms of elliptic curve

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?
1
vote
2answers
52 views

Prove the nonexistence of $p$-torsion for $p > 3$ in $E:y^2 = x^3 + ax$ for prime $a \geq 2$.

$$\Large\textbf{Problem}$$ Let $E$ be an elliptic curve defined by $y^2 = x^3 + ax$ where $a \in \mathbb{Z}$ is fourth-power free. Then \begin{aligned} E(\mathbb{Q})^{\text{tor}} = \left\{ ...
3
votes
1answer
78 views

Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
2
votes
1answer
76 views

Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
0
votes
1answer
75 views

Trivial torsion subgroup

I am just wondering, suppose we have a curve $y^2 = x^3+ax + b$ defined over $\mathbb{Q}$ and suppose for simplicity $a,b \in \mathbb{Z}$. Can we say something about the torsion subgroup with the only ...
1
vote
0answers
66 views

Elliptic curves in $\Bbb P^3$

How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by ...
4
votes
2answers
69 views

Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...
4
votes
1answer
112 views

Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
0
votes
0answers
27 views

calculate canonical height ellyptic curve

Hello everyone this is my first post here and this is probably an easy question, but i am not a math geek.... I need to calculate the canonical height af a Point on an ellyptic curve in python.... I ...