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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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A strong form of the Nagell-Lutz theorem

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ for $a,b∈ℤ$. Let $O$ be the identity element of $C(ℚ)$. I know…
Safwane
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Another Isomorphism between elliptic curves

$E/\mathbb{C}$ and $E'/\mathbb{C}$ are isomorphic elliptic curves. Then if $$E :\ y^{2} = x^3 + Ax + B$$ then $$E': \ y^{2} = x^3 + \mu ^4 Ax + \mu ^6 B$$ and the isomorphism map $\phi : E \to E'$ is $$\phi (x, y) = (\mu^2x,…
alpha
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Point at infinity on quartic elliptic curve

Elliptic curve defined by $$E_1: y^2=7 x^4+x^3+x^2+x+3, P_1=(-1,3)$$ can be transformed to $$E_2: v^2=u^3-\frac{250 u}{3}-\frac{1249}{27}$$ Substitutions used are: $$\left(x\to \frac{15 u-9 v+217}{39 u+9 v+209},y\to \frac{9 \left(54 u^3+639 u^2-27…
azerbajdzan
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Checking whether a point is on an elliptic curve

I have the point $(2,1)$ and I would like to check whether it is on my elliptic curve. The elliptic curve is defined by $y^2 = x^3 + 3x + 1$ in $GF(7)$. My solution to check whether or not the point is on the curve was to substitute the point into…
user3371137
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From a quartic to a cubic in Weierstrass form

I have a quartic in $(u,v): v^2=u^4+au^2+b$ and I want to transform it in a cubic form in $(X,Y)$ and then in a cubic Weierstrass form in $(x,y)$. I know by the Nagell algorithm, that I have to introduce $G(u)=u^2+a/2$, so that I can…
beginarray
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Isogeny between j-invariants

I am studying $l$-isogeny graphs (volcanoes). As I understand these graphs have $j$-invariants as vertices but I am having a hard time understanding the edges. The following is not clear to me: Suppose $E_1/\mathbb{F}_q, E_2/\mathbb{F}_q$ are…
sugyman
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elliptic curve ${X^3+Y^3=AZ^3}$

consider the elliptic curve $X^3+Y^3=AZ^3$ and $A$ in $K*$ with $O=(1,-1,0)$. show the $j$ invariant of this elliptic curves is $0$. (part d of Silverman exercise page 104 Q3.3 ) I can compute the $j$ invariant of Weierstrass form of elliptic…
bluemath
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Descent by 2-isogeny

Im practicing with a exercise about 2-isogenies, but struggling a bit. Im doing the following exercise: Given two elliptic curves over $\mathbb Q$. \begin{equation}E: y^2 = x(x^2-5) \quad E':y^2 = x(x^2+20)\end{equation} These are related by a…
Banjobill27
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Elliptic curves over $\mathbb{Q}(\sqrt{2})$ and maps between them

Let $K = \mathbb{Q}(\sqrt{ 2})$. Let $E_1$ be an elliptic curve over $K$. What is the structure of the set of points $E_1(K)$? Assume there is an elliptic curve $E_2$ defined over $K$ and a map $f : E_1 \rightarrow E_2$ defined over $K$ that is…
user289143
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Examples of non-minimal Weierstrass equations that make $[E(K):E_0(K)]$ arbitrarily large.

I'm taking a course on elliptic curves, in which the lecturer briefly mentioned that the Tamagawa number $c_K(E)=[E(K):E_0(K)]$ satisfies $c_K(E)=\mbox{ord}(\Delta)$ or $c_K(E) \leq 4$. He said that this is only true if we use a minimal Weierstrass…
porkramen
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What is the area of elliptic curve determined by the metric at cotangent space?

This question may be too low-level even for this site. For simplicity let us assume the elliptic curve to be $\mathbb{C}/\{1, \tau\}$ where $\tau=x+iy$. Let the metric on the Riemann surface to be given by $$ |dz|=C $$ My question is, what is the…
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The Group of points on the Elliptic curve $y^2=x^3+1$ over $\mathbb{F}_5$

So I'm trying to understand the group of points of $y^2=x^3+1$ over $\mathbb{F}_5$ and for some reason I seem to be getting nonsense answers and I'm not sure what I'm doing wrong. So basically my formulas for this particular elliptic curve…
Set
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Halving a point on an elliptic curve

I have come across a few scholarly articles on halving points on an elliptical curve (https://eprint.iacr.org/2011/461.pdf, for example) but it doesn't work with my curve. Can anyone explain to me how to halve a point on the elliptical curve y^2 =…
Tom V
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EC Point calculation

Following the Guide to Elliptic Curve Cryptography, it provides the following elliptic curve on $E(\mathbb{F}_p)$ with $p=29$ on page 80: $E: y^2 = x^3 + 4x + 20$ Page 81 provides a list of the points on the curve. For $x=2$ it includes the points…
Dennis
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Two-Torsion of Elliptic Curves and Their Twists

Let $E/\mathbb{Q}$ be an elliptic curve such that $E(\mathbb{Q})[2]=\{\mathcal{O}\}$. Does there exist a quadratic twist $E^{(d)}$ of $E$ with a $\mathbb{Q}$-rational two torsion point? I'm particularly interested in the case where $E$ has complex…
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