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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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Order of an equation on an elliptic curve

I have the following statements: For $E$, an elliptic curve over $\mathbb{F}_{11}$ given by $y^2=x^3+4x$, I have: 1a. The line through $(0,0)$ and $(2,4)$ as $y-2x=0$ and am told that its order is $2$ For $E$, an elliptic curve over $\mathbb{F}_7$…
lioness99a
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Elliptic Curve and Divisor Example help (Step 2)

I have an elliptic curve $E$ over $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ with the point at infinity $\mathcal{O}$ I have a divisor of $E$, defined by…
lioness99a
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Elliptic curves of degree $d$ in $\mathbb{P}^n$

I have the following question on elliptic curves immersed in the projective space $\mathbb{P}^n$. More precisey: let's take $\tau \in \mathcal{H}_1$ an element in the upper half-plane in $\mathbb{C}$ and the elliptic curve $E_{\tau}$ given by the…
Lucke
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Elliptic surface calculator

In kuwata's package : elliptic surface calculator i want to transform quartic to cubic with specific point but every time it say's : the point is not on the curve . can anybody tell me the correct command? maybe it have some problem . i asked this…
user371596
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Plotting cubic curves

I wanted to understand how does one plot graphs for cubic functions. for example , $y^2 = x^3 + 2x + 10$ Since this is a cubic function , i understand for every x value , y has a reflection along the x axis. Question being how does one compute the…
Bobo
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Is it true that $x^3 + a_4 x + a_6$ is a perfect square 50% of the time?

Can it be said that $$x^3 + a_4 x + a_6 \; (\text{mod}\,p)$$ is a perfect square half of the time? Apparently it does not. Simulating the elliptic curve points $\text{mod}\,103217$ only $310$ of them are perfect squares. >>> from gmpy2 import…
Giuseppe Crinò
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Order of an Elliptic curve

Kindly asking for any hints about the following question: Let $E$ be an elliptic curve over $F_p$ where $p>7$ is a prime. Suppose $E(F_p)$ had a point of order $p$, then if #$E(F_p)=p$?
Masoud
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Isomorphism between elliptic curves

If $2$ elliptic curves over a field with characteristic different from $2$ and $3$ are isomorphic via an invertible linear map $\mathbb P^2\to\mathbb P^2$, then how does this map look like ? (the task is to prove that their $j$-invariant is equal,…
user257
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Is there any Elliptic Curve algorithm equivalent to RSA's asymmetric encryption?

I've been searching but I cant find anything about this... only EC Diffie-Hellman with symmetric cryptography, which is exactly what I do not want :( Imagine this: generate a random private key, k then calculate a "public key" with ECC, d=kG The…
WhyWhat
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Why the proof of III 7.4 of J. Silverman's AEC is not way easier?

In Chapter 3 of "Arithmetic of Elliptic Curves" Theorem 7.4 says: given $E_{1}, E_{2}$ elliptic curves, the map: $Hom(E_{1},E_{2})\otimes \mathbb{Z}_{l}\to Hom(T_{l}(E_{1}),T_{l}(E_{2}))$ obviously defined, so that $\Phi\mapsto \Phi_{l}$ is…
Hair80
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Silverman Arithmetic of Elliptic Curves Exercise 3.5(a)(ii)

As the title says, this question is about the final part of Exercise 3.5(a)(ii). Let $E/K$ be given by a singular Weierstrass equation $f(x,y)$. Suppose that $E$ has a node, and let the tangent lines at the node be $y=\alpha_1+\beta_1$ and…
CurveEnthusiast
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What is the kernel polynomial of a degree $l$ isogeny?

Given an elliptic curve $E$ in Weierstrass form. Is there a standard way to describe the kernel of a degree $l$ separable isogeny? More specifically, can the kernel polynomial be written down, only knowing $E$ and that the isogeny is separable and…
s3binator
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Finding torsion subgroups of elliptic curves over finite fields

Finding torsion subgroups of elliptic curves over finite fields. Given $y^2=x^3+x+1$ over $F_3$ I need torsion subgroup of $E[3]$ $E[3]$ is either trivial or isomorphic to $\mathbb Z_3$ The points $(1,0),(-1,0),(0,0)$ are each of order $2$, so…
user257
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Intuition for modular curves and their connection to elliptic curves

Using Diamond, I have read and understand the definitions of modular curves as quotient groups of the upper half plane, modded out by congruence sub groups of $SL_2(Z)$, I understand that they can be interpreted as Riemann surfaces, then…
s3binator
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How to prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$

Let $f(x) = x^3 +ax+b$ contained in $Q[x]$ prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$ This is what I've so far since $f(x) = (x-A_1)(x-A_2)(x-A_3), A_1,A_2, A_3$ are roots of $f(x)$ and $discriminant(f) =…
etet112
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