Mathematics Stack Exchange
Mathematics Stack Exchange

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 .

3283 questions
0
votes
0 answers

Completion of proof of theorem 4.1 in Husemöller Elliptic curves

Edit : The proof of what I'm asking can be found in Anthony Knapp's book Elliptic curves page 85. Husemöller's book : On page 38 of the second edition and on page 37 of the first, you can find the following theorem (4.1) Theorem : Let $E$ be an…
Identity
  • 1
  • 2
0
votes
1 answer

About the separable maps of elliptic curves

We know that when $\alpha \neq 0$ is separable, $\deg(\alpha) = \#E(F_q)$. Also the Frobenius map $ \phi_q (x,y) = (x^q,y^q)$ is not separable and it has degree $q$ and the map $1-\phi_q = (x-x^q,y-y^q)$ is separable and has degree $q$. But this…
CCCC
  • 167
0
votes
0 answers

Verifying $\ker( \phi_q -1) = E(F_q) $ by an example

I want to verify $\ker(1- \phi_q ) = E(F_q) $ by an example where $\phi_q$ is the Frobenius map $(x^q,y^q)$. If we take $E: y^2= x^3 + 3x +2$ over $F_5$, then the points on it are ${(1,1),(1,4),(2,1),(2,3),(4,0),\infty}$. But I do not see how these…
CCCC
  • 167
0
votes
1 answer

How to get $y_1$ and $y_2$ values from $y^2=x^3+ax+b$ in elliptic curve

I am studying Elliptic curve and I am trying to solve the $y_1$ and $y_2$ values as in this document: https://www.site.uottawa.ca/~chouinar/Handout_CSI4138_ECC_2002.pdf As in Page 2 I can find $y^2$ but in some points I can not understand why when…
Cisco Saeed
  • 109
0
votes
1 answer

constant term of division polynomials of elliptic curves

I am mainly looking for a reference request, if it exists, to save some time looking for it. Let $E/\mathbb Q$ an elliptic curve in Short Weierstrass form and $\phi_n,\omega_n,\psi_n$ the usual division polynomials ( $nP=(…
fhn
  • 1,097
0
votes
1 answer

Given an order $\mathscr{O}$of imaginary quadratic field, is there an elliptic curve $E/\mathbb{C}$ s.t. $\operatorname{End}(E)\cong \mathscr{O}$?

Sorry for my bad English. For a CM elliptic curve $E/\mathbb{C}$, the endomorphism ring $\operatorname{End}(E)$ is isomorphic to an order of an imaginary quadratic field. I want to know if it is true this inverse proposition.
Yos
  • 1,924
  • 7
  • 14
0
votes
1 answer

What is polynomial $f(X)\in \mathbb{F}_{p^2}$ s.t. $f(j)=0$ iff $j$ is supersingular.

Sorry for my bad English. Let $p$ be a prime. We say $j\in \mathbb{F}_{p^2}$ is supersingular if the corresponding elliptic curve to $j$ is supersingular. So there is polynomial $f(X)\in \mathbb{F}_{p^2}[X]$ s.t. $f(j)=0$ iff $j$ is supersingular. I…
Yos
  • 1,924
  • 7
  • 14
0
votes
2 answers

Any general algorithm to find $x$ given $y$ for the elliptic curve equation $y^2 = x^3 + ax + b$, in real numbers?

Is there a way to solve for $x$ given $y$, in the equation $y^2 = x^3 + ax + b$, in the real number space (not finite field). Or is the solution too difficult to achieve? I suspect the solution will require factoring a 3rd degree polynomial. I'm…
Sal Rahman
  • 244
0
votes
0 answers

Models of an Elliptic Curve

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then it has a model $W$ satisfying $W(\mathbb{Z})=E(\mathbb{Q})$. This means $E$ only has integer points. But this is not true. What went wrong?
Mikko Pitkonen
  • 55
0
votes
0 answers

How to translate $n=x^3+y^3$ into Weierstraß-form $y^2=x^3+Ax+B$?

I am currently dealing with the question of which numbers can be expressed as a sum of two rational cubes ($n=x^3+y^3$, $x,y\in\mathbb{Q}$). I've already learned that I can understand the equation $n=x^3+y^3$ as an elliptic curve (given by…
känguri
  • 1
0
votes
1 answer

Evaluation of a rational function in the elliptic curve unit element

I'm reading on divisors and elliptic curve pairing. For a field $F$ and a rational function $f(x,y) \in F(x,y)$ it's often written $f(P)$ for points $P$ on the curve. But what is $f(P)$ when $P = \infty$, i.e. the unit element for elliptic curve…
kuco 23
  • 153
  • 5
0
votes
2 answers

Where is the "point of infinity" for elliptic curve over finite fields?

This might be a dumb question, but I can't find the answer. Everywhere I see people say elliptic curve is this set of points plus a point of infinity on a finite field. Where is that point of infinity? Because finite field is finite, is the point of…
Justin Zhang
  • 113
0
votes
1 answer

Bringing an Elliptic curve in homogenous form to Weierstrass form

I have a family of curves given by $F(U,V,W)= U^3 +V^3 + W^3- 3\lambda UVW$ in $\mathbb{P^2C}$, with an origin $O = [1,-1,0]$. I am struggling to bring this to the form $y^2 = x^3 - ax +b$ as I can't seem to change the basis correctly. I have the…
Abu Nazir
  • 1
0
votes
0 answers

A question on the $2$-torsion points of a elliptic curve

I am reading Saito's Fermat's Last Theorem: Basic tools and on page 15, it was claimed that Proposition 1.4. Let $K$ be a field with char($K$) $\neq 2$, and let E be an elliptic curve over $K$. Then all $2$-torsion points of $E$ are $K$ rational if…
wilsonw
  • 1,016
0
votes
1 answer

Is there a relationship between points on 2 different elliptic curves that share zeros?

FYI: I'm new to elliptic curves but want to learn and understand. I would like to work with the elliptic curve $C_1: y^2=x(x+1)(2x-1)$ over $\mathbb{Q}$ but I'm having trouble finding rational points other than the roots. Can I use a root, say…
pgpilot326
  • 25
1 2 3
…
26 27