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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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Is a twisted Edwards curve an elliptic curve?

I don't belong to the Math department. Currently, I am studying Elliptic curves for cryptographic applications. While going through the definition of elliptic curves, it states that Elliptic curve is a curve of the form $y^2 = p(x)$, where $p(x)$…
viji
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Distribution the points in Elliptic curves over a finite field F_p where p is prime.

I want to know why the distribution the points in Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime is uniform. That means the number of points in elliptic curve $E$ with $x$-coordinate in the interval…
user53917
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Elliptic Curve Group Cycle At Inflection Point?

The elliptic curve $ y^2 = x^3 -x + b $ has 2 points of inflection where $ y'' = 0 $. Visualized here. It seems that $ P + P $ for at say the upper point would be $ - P $ since the tangent at $ P $ intersects "triply" at $ P $ and therefore $ P + P…
user2297550
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Fields of definition for torsion points on elliptic curves over finite fields

This is about the following proposition from R. Schoof's article Nonsingular Plane Cubic Curves over Finite Fields: Proposition 3.7: Let $E$ be an elliptic curve over the finite field ${\mathbb F}_q$ of characteristic $p$ and $n\in {\mathbb N}$…
Hanno
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Elliptic Curve addition: Tangent Point

Considering the "visual" method of explaining elliptic curve addition, I wonder what you are supposed to do if one of the points being added is tangent to the curve? With the classic shape e.g. $y^2 = x^3 - x + 2$ , you could get a horizontal line,…
T. Hrowaway
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Eisenstein series --

I need some help for the proof of the uniformization theorem (Silverman's Advanced Topics ...). If we have $G_{4}(\Lambda_{1})=G_{4}(\Lambda_{2}) $ and $ G_{6}(\Lambda_{1})=G_{6}(\Lambda_{2})$ (with $\Lambda_{1},\Lambda_{2}$ two lattices and…
doeup
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genus of an elliptic curve

I'm trying to show that the curve $f(z,w)=z^4-w^2+1$ has genus 1. The curve is clearly non-singular, so I tried using the degree formula \begin{equation*} g=\frac{(d-1)(d-2)}{2}=\frac{3\cdot2}{2}=3, \end{equation*} but it should be equal to one. I'm…
mj_indefinite
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Torsion points on elliptic curve

Please help or hints me to solve this question: Suppose that $E: y^2+y= x^3+x$ be an elliptic curve over $\Bbb F_2$, We know that $E$ is supersingular. Show that $E[5] ‎‎\subseteq‎‎ E(\Bbb F_{16})$.
Masoud
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Ranks of elliptic curves as the base field varies

Suppose that $E/\mathbb{Q}$. Is it the case that for any $r > 0$ there exists a finite field extension $\mathbb{Q} \subset K$ such that the rank of $E/K$ is greater than $r$?
Jonah Sinick
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Elliptic curve condition on coefficients

I am working something where a picture like this one appeared : Say the curve is written in the form $$ y^2 = x^3 + ax^2 + bx + c $$ (if this is the wrong form of coefficients, feel free to correct me, I am guessing here) what are the conditions…
Patrick Da Silva
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Group law on Elliptic Curves over $\mathbb F_5$

I want to find the group table of the elliptic curve $\mathcal C : y^2 = x^3-x$, defined over $\mathbb F_5$. I get the points on $\mathcal C$ to be the identity ${\bf o}$, together with $(0,0)$, $(1,0)$, $(2,1)$, $(2,4)$, $(3,2)$, $(3,3)$ and…
Fly by Night
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Reduction of endomorphism ring of elliptic curve

Let $E$ be an elliptic curve defined over a number field without complex multiplication and with ordinary reduction at a prime $p\in\mathbb{N}$. When is the reduction mod $p$ map a surjection on the endomorphism ring I.e. $\overline{End(E)} \cong…
Peter
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Order of a point on elliptic curve

I am trying to prove that the following elliptic curve has rank=1: $$y^2=x^3+x^2+x+1$$ From the map $$\delta: E(Q)\rightarrow Q(i)^*/(Q(i)^*)^2$$ $$(x,y)\mapsto (x-i) $$ for $(x,y)\neq O$ and $O\mapsto 1$, I can show that $E(Q)/2E(Q)\cong…
kimtuan
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sum of torsion of an elliptic curve

It is clear from the isomorphism between elliptic curves over $\mathbb{C}$ and complex tori that the sum of the $m$-torsion points is the identity in the group law of the elliptic curve. How generally does this hold, and how can one see it (not…
Tony
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If $\phi$ is an endomorphism of an elliptic curve, and $\phi = \hat{\phi}$ then $\phi = [m]$?

I heard a reference to this fact, but I cannot find a reference. (I can find the converse in Silverman, namely that $\hat{[m]} = [m]$.) Notation: $[m]$ is multiplication by $m$ in the group law, and $\hat{\phi}$ is the dual endomorphism of…
Elle Najt
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