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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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How come $f(0) = 0$ in $\mathbb C/L$?

How come $f(0) = 0$ in $\mathbb C/L$? Does anyone know it? Your help will be appreciated. This is taken from the text "Rational Points on Elliptic Curves" by Tate and Silverman.
Bobby
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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 $f(x,y)=y^2-x^3-ax-b$. Then I tried to calculate there $\nabla…
unterbrause
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Weierstrass normal form

How can I show that the Weierstrass normal form $u^3 + v^3 = \alpha$,with $x=12\alpha/(u+v)$ and $y=36\alpha (u-v)/(u+v)$, satisfy $y^2=x^3-432α^2$ ?
Alex
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Weierstrass to general form

Can we go from short Weierstrass equation equation $y^2=x^3+Ax^2+Bx+C$ to general $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$?
user119081
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Calculating Non-Singular Map of Elliptic Curve

I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity to infinity. I've read a lot about the steps I'm…
user3081739
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isogenies between tori

Let Hom$(\mathbb{C}/\Lambda_1,\mathbb{C}/\Lambda_2)$ be the set of isogenies between $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$, where $\Lambda_1,\Lambda_2$ are lattices. I am asked to prove what the structure of this group is…
user100659
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Find the bound for [K(E[p]):K]

Let E be an elliptic curve over a field K of characteristic p > 0, we know that E[p] has order 1 or p, how to bound [K(E[p]):K]?
Yang Wesley Yang
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Weierstrass form for some equation

How to find a birational transformation that turns the equation $3(y^2-1)=2x^2(x^2-1)$ into Weierstrass form? Thanks!
Yang Wesley Yang
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Why is $p$ (or $p + 1$) not the upper bound of the number of solutions of an elliptic curve mod $p$ (finite field)?

In the build-up to the enunciation of the Birch, Swinnerton Dyer conjecture (BSD), the following back-of-the-envelop idea comes up: Because half of the $1$ to $n-1$ elements mod $p$ are quadratic residues (squares), for the expression $y^2= x^3…
JAP
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Points on Hyperelliptic Curve

Here: https://en.wikipedia.org/wiki/Imaginary_hyperelliptic_curve#The_divisor_and_the_Jacobian it says: "Actually, Bézout's theorem states that a straight line and a hyperelliptic curve of genus $2$ intersect in $5$ points. So, a straight line…
just_learning
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Determinant of elliptic curve torsion subgroup

In Silverman's Arithmetic of Elliptic Curves page 93, he defines $$\det:E[m]\to \mathbb{Z}/m\mathbb{Z}, \quad \det(aT_1+bT_2,cT_1+dT_2)=ad-bc$$ where $\{T_1,T_2\}$ is a basis for $E[m]$. I don't get why elements in $E[m]$ are of the form…
PatrickStewart
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Inflection points on elliptic curves

I'm looking at how to transform a cubic into a Weierstrass equation on page 52 of this elliptic curves pdf here. The author writes: “There are two distinct transformations depending on whether the point $P$ is an inflection point on $C$ or not.” On…
Peter4075
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How to find 3P of Elliptic curve when slope is a decimal?

Hi I was soliving Elliptic Curves and I found out $2P$ but when it came to $3P$ I became a little bit stumped. My equation is $$y^2 =x^3 + 2x + 9 \qquad(\text{mod } 23)\ .$$ The values I got for $P$ is $(0,3)$ and $2P$ is $(18,14)$, I do know that…
Eric Varghese
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Map between Points of two different Elliptic Curves of same order

Let $E_1$ and $E_2$ be two elliptic curves defined over $F_{p1}$ and $F_{p2}$ respectively and with same number of points $\#E_1(F_{p1}) = \#E_2(F_{p2})$. Since number of points of two curves are same, then there should exist one to one map between…
AlphaCentauri
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Why an R-valued point on an arithmetic surface can be specialized to the generic fiber?

Let $R$ be a Dedekind domain with $K=Frac(K)$, $\mathcal{C}$/$R$ be an arithmetic surface, and let $C/K$ be the generic fiber of $\mathcal{C}$. Why any point in $\mathcal{C}(R)$ can be specialized to the generic fiber to a point in $C(K)$? In fact,…
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