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

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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 ...
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1answer
49 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 : ...
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1answer
40 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 ...
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0answers
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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 ...
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1answer
28 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 $E(F_q)[n]$ ...
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23 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, ...
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2answers
69 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, ...
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3answers
93 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 ...
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1answer
23 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
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1answer
138 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, ...
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1answer
49 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 ...
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4answers
136 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 ...
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0answers
36 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
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1answer
45 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 ...
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1answer
120 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.
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27 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?
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1answer
34 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 ...
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1answer
82 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 ...
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1answer
49 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 $b$ ...
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1answer
82 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 ...
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1answer
34 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. ...
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0answers
90 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 ...
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2answers
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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 ...
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1answer
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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?
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2answers
49 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\{ ...
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73 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 ...
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1answer
72 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 ...
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1answer
67 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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1answer
112 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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0answers
60 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 ...
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25 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 ...
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1answer
94 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 ...
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1answer
128 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 = ...
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1answer
124 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 ...
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1answer
81 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 ...
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1answer
74 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 $p$"?
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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 ...
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89 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 ...
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1answer
31 views

Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...
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1answer
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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 ...
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2answers
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Geometric picture of 3-torsion points on an elliptic curve

I'm faced with what seems a paradox. If we have an elliptic curve $E/\Bbb C$ in Weierstrass form so that $\mathcal O_E$ is at infinity, then the addition law is quite easy to picture geometrically. In ...
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1answer
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Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero

Silverman, p. 104: Show that the polynomials $$f=x^4−b_4x^2−2b_6x−b8 \qquad \text{and}\qquad g=4x^3+b_2x^2+2b_4x+b_6$$ appearing in the duplication formula (III.2.3d) are relatively prime if ...
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1answer
72 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
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2answers
118 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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1answer
30 views

Why does an isogeny not ramify?

The following argument is, I believe, based on the premise that an isogeny (or a morphism of curves that is a group homomorphism) doesn't ramify: Considering the multiplication by $n$ map $[n]$ on a ...
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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 ...
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Local coordinate of a (hyper)elliptic curve at infinity

I would like to ask for some help to clarify the following: In (9) of page 7 of http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf one finds stated that $(\mu, \lambda) \mapsto ...
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1answer
77 views

Odd torsion of elliptic curves are isomorphic

$C: Y^2=X(X^2+aX+b)$ $D: Y^2=X(X^2+a_1X+b_1)$ where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$ Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which ...
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1answer
53 views

Finding J-invariant of Legendre form of Elliptic Curve

PROBLEM: Put the Legendre equation $y^2 = x(x − 1)(x − λ)$ into Weierstrass form and use this to show that the j-invariant is j = $2^8\frac{(λ2 − λ + 1)^3}{λ^2(λ − 1)^2}$ . Recall: Weierstrass ...
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1answer
37 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...