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

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7
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2answers
92 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
35 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
18 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
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1answer
24 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
19 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
40 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
votes
0answers
23 views

Find point of maximal order $\operatorname{lcm}(a,b)$ on a curve

In Elliptic curve okamoto uchiyama there is a condition for picking base point $G$ such that $G$ belongs to $E_n$ of maximal order $\operatorname{lcm}(|E_{p2}|,|E_q|)$. I am not getting these ...
1
vote
1answer
25 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
13 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, ...
1
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2answers
24 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
68 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
20 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 ...
1
vote
1answer
36 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
35 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
84 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
174 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
56 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
35 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 ...
8
votes
4answers
103 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
43 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
32 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 ...
3
votes
1answer
104 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
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0answers
23 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
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1answer
17 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
78 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
35 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
81 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
30 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
43 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
175 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 ...
7
votes
2answers
54 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
83 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
51 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
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2answers
40 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
57 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 ...
4
votes
1answer
88 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 ...
2
votes
1answer
61 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
54 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 ...
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0answers
56 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
63 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
79 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 ...
1
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0answers
23 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 ...
6
votes
2answers
69 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
1
vote
1answer
73 views

Finding points on an elliptic curve

I have an elliptic curve $$x^3+17x+5 \mod 59$$ $P = (4,14)$ is given and I need to find point $8P$. to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = ...
2
votes
1answer
69 views

Order of a point on an Elliptic Curve

I am currently struggling with the determination the order of a point on an elliptic curve. We had to do the following exercise: $C = V(y^2+x^3-1)$ and $P = (0,1)$. Now Wikipedia told me that I can ...
3
votes
1answer
54 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
3
votes
1answer
51 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
25
votes
0answers
253 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
1
vote
0answers
21 views

Rationality of divisors at infinity

In an attempt to clarify to myself some terminology (ant the scope of the Riemann-Roch theorem), I would like to ask for examples of genus $1$ curves of the form $$C : y^2 = ax^4 + b$$ where $a, b \in ...
0
votes
1answer
41 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...