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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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What is the right dual isogeny?

I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of unity. Then the elliptic curve $C: y^2=x^3+1$ has…
TheBeiram
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Finding formula for rational point ($x$,$y$) on the y = $x^2$

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))
Muneeb
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Properties of the elliptic curve $y^2 \equiv x^3 – 2 \pmod 7$

Can someone help me: 1) to list the points on the elliptic curve $E: y^2\equiv x^3 – 2\pmod 7$. 2) to find the sum $(3, 2) + (5, 5) $ on $E$.
Niyaz Iralin
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How to pass from general weierstrass equation of elliptic curves to shorter ones

I found a page on wikipedia about weierstrass equations of elliptic curves. https://fr.wikipedia.org/wiki/%C3%89quation_de_Weierstrass The page says that one can put weierstrass equations in shorter forms depending on the characteristic of the…
user28082
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number of points of order 2 on an elliptic curve

Given a field $F$ with $char F \neq 2,3$ and an elliptic curve $E: y^2-(x^3+ax+b)$ I want to find the number of points of order $2$. (The given solutions say it is always exactly 3.) Let $P$ be a point of order $2$: If $P\oplus P=0$ then $P=-P$…
flawr
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Dual and degree of the isogeny a+b[i].

Let $[i]$ be the endomorphism such that $[i](x,y) = (-x,iy)$ with $[i]^2+[1]=[0]$. I am trying to prove that the degree of the endomorphism $[a]+[b]\circ[i]$ is equal to $a^2+b^2$. After multiplication in $End(E)$ by $[a]+[-b]\circ[i]$ I get…
user288451
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Elliptic curve of points order 4

So how do you find the points on the eliptic curve $y^2=x^3+ax$ of order 4, where $4\mid a$ but $4^n$ does not divide $a$ for $n>1$. We proved that for $(x,y)=2(u,v)$, we must have $x=(u^2-a)^2/(4v^2)$. Now I wish to prove that if points $P$ with…
nerd
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Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or at least I have calculated, hopefully correctly)…
Noxbru
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Dividing elliptic curve point on integer

Can we solve the equation $nP = Q$, where $P$, $Q$ is a rational poins on elliptic curve ($P$ is unknown), and $n$ is integer?
Michael Galuza
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Group on elliptic curve points

Let's we have an elliptic curve (EC). Is it possible to construct group $G$ acting on the points of an EC with this property: if $P$ is a rational point on EC then $G(P)$ is also is a rational point? Of course, I means that $G$ is not reduced to…
Michael Galuza
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Sato-tate conjecture for elliptic curves over finite fields

I am doing a research project about elliptic curves over finite fields and I am across the Sato-tate conjecture, but I am having some difficulty understanding it. What I (think) I took away from the reading: it states how the error term ($2\sqrt…
Math Major
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Chinese Remainder theorem on Elliptic Curve group

I read somewhere (Blake, Seroussi, Smart: Elliptic Curves in Cryptography, p.160) that one can use the Chinese Remainder theorem to split $E(\mathbb{Z}/N\mathbb{Z})$, where $N$ is a composite number. Let me set up the question: Let $N=pq$, where…
BlackAdder
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Multiplication by n on E(K) is surjective

What's the easiest way to see this? I can imagine a proof for $n=2^k$ since for some $P \in E(K)$ you can just move a line intersecting P round the curve till it's tangent, then that point, say $Q \in E(K)$ would be such that $2Q=P$ and induction…
user70147
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Was the Wiles's proof of FLT based on elliptic curves or generalized elliptic curves?

I have been told that Wiles's proof of FLT was based on elliptic curves. But yesterday I read from Takeshi Saito's book "Fermat's Last Theorem Basic Tools" that there is so called generalized elliptic curve, kind of proper flat scheme and whose…
mathenthusiastic
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Elliptic Curves

I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is nonzero and Tate Shafarevich group is also nontrivial…
MKJ
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