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Questions tagged [elliptic-curves]

For questions about elliptic curves.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$, defined over any field $K$. They form abelian groups under point addition. They are much studied in number theory, for example in cryptography and integer factorization.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=-16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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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 four other integer points $P_4(-508, 262128)$,…
Dan Ismailescu
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Clarification of the proof of **Proposition 1.5** in Silverman

I'm studying the proof of Proposition 1.5 on silverman "The Arithmetic of Elliptic Curves" : Proposition: Let $E$ be an elliptic curve. Then the invariant differential $\omega$ associated to a Weierstrass equation for $E$ is holomorphic and…
Med
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Generators of elliptic curves?

How to find generators to group $E(\mathbb Q) $ of following elliptic curves $E:y^2=x^3-198 $, $E:y^2=x^3-122 $. Thank you in advance.
MKJ
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$\forall p\geq 3, E:y^2=x^3+x$ satisfies $\#E(\mathbb{F}_p)=0\mod4$

The question is taking from Arithmetic of Elliptic Curves by Silverman, Q5.12 on page 154. I've managed to show the supersingular case when $p=3 \mod 4$, which was done more generally for elliptic curves of the form $y^2=x^3+ax$. I'm now trying to…
Haikal Yeo
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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 quoting the Selmer groups? Well, a classical approach on…
user137857
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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 particular, the 2-torsion points are exactly the…
Rodrigo
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Birational equivalence of cubic with a Weierstrass form

I want to convert the cubic $x^3 + 3y^3 - 11z^3 = 0$ to Weierstrass form (to find its rank) so I tried to follow the suggestion from Timothy: I found three points $(2:1:1),(28:-19:5),(-537656:443213:212645)$ on the curve and used a linear change of…
user16697
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Zeros and poles of rational functions over elliptic curve

On this page, the author states: It turns out this definition can be extended to points of order 2, and also the point O (when we homogenize the functions and work over the projective plane). Moreover, every rational function has as many zeroes as…
ted.k
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Rational points of order 2 on elliptic curves

I'm working on a question that's asking me to list all the $\mathbb Q$-rational points of order 2 and all the $\mathbb C$-rational points of order 2 for some elliptic curves. I've made the following observation but can't be sure whether I'm…
Haikal Yeo
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Frobenius Endomorphism

I had a lecture last week which dealt with the Frobenius Endomorphism on elliptic curves. The lecturer showed an example at the end of the lecture, when almost out of time and I don't quite understand it. We had the elliptic curve $E: Y^2+XY=X^3+X$…
user106421
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Quadratic twist of an elliptic curve

I found this page: http://en.wikipedia.org/wiki/Twists_of_curves#Quadratic_twist which tells me $dy^2=x^3+a_2x^2+a_4x+a_6$ is equivalent to $y^2=x^3+da_2x^2+d^2a_4x+d^3a_6$. Why is this equivalent (for $d$ as given on that page)?
user100659
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Division polynomials of elliptic curves being squares

Let $E$ be an elliptic curve over a field $K$ given by a long Weierstrass equation $e \in K[X, Y]$, and let $\psi_n \in K[X, Y]$ be the $n$-th division polynomials of $E$. Can we formally prove that $-\psi_{n + 1}\psi_{n - 1}$ is a square in the…
Multramate
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Why the author choose $s$ real?

My question is: Why the author of this book (http://wstein.org/books/bsd/bsd.pdf) page 8 on Sec 1.4 choose $s$ real in despite that the variable is complex in the entire chapter. I am very confused about this fact.
Safwane
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What is a "universal algebraic curve of genus 1"?

I need to understand how exactly the construction of the universal algebraic curve (in this sense) goes in the context of elliptic curves over the complex numbers.
Matt Park
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Fermat curve in Weierstrass form.

In this post (Substitutions that transform Fermat Equations to Elliptic Curves) it is proved that there exists a change of variables that trasform Fermat's curve $X^3+Y^3+Z^3=0$ into $y^2=x^3-432$, which is an elliptic curve in Weierstrass form. But…
Marcos
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