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
6
votes
1 answer

First problem in Silverman's Arithmetic of Elliptic Curves

I started working through Silverman's Arithmetic of Elliptic Curves. For some reason it looks like the first problem in the first chapter is the hardest problem in the whole chapter or I'm completely missing something. The Problem statement is the…
JT1
  • 653
6
votes
2 answers

How to find positive rational points for both x and y in $x^3+y^3=1141$?

I did some research about finding a rational solution for this equation $$ x^3 + y^3 = 1141\ , $$ and learned a little about elliptic curves. However, those solutions require a known rational solution. How do you solve this without a known rational…
Burgger Big
  • 63
  • 4
6
votes
1 answer

Number of 3-torsion points on an elliptic curve

If we take our elliptic curve over $K$ to be the zero set of $$ F(X_1, X_2, X_3) = X_2^2 X_3 - (X_1^3 + AX_1X_3^2 + BX_3^3), $$ which is in projective form with $X = X_1, Y = X_2, Z=X_3$, then I have been able to show that for any point $P$ on the…
Sputnik
  • 3,764
6
votes
2 answers

Point addition on an elliptic curve

I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$. I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive that answer. I have $6\cdot(5, 1) = (16,13)$ and…
Peanut
  • 205
6
votes
0 answers

Help with showing that $\text{Gal}\big(\mathbb{Q}(i)(E[n])/\mathbb{Q}\big)$ is abelian

Since it is my first post, I want to say hello to everyone :) I have a problem with exercise 17c from the book "Rational points on elliptic curves" by Silverman and Tate. I have given the elliptic curve $E: y^2=x^3+x$ and $K_n=\mathbb{Q}(i)(E[n])$…
filipux
  • 61
6
votes
2 answers

Rank of the elliptic curve $y^2=x^3+px$

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an integer solution in M, N and e (with M, N and e…
Shiva
  • 561
6
votes
5 answers

upper bound on rank of elliptic curve $y^{2} =x^{3} + Ax^{2} +Bx$

I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies $r \leq \nu (A^{2} -4B) +\nu(B) -1$ where $\nu(n)$ is the number of distinct…
pel
  • 1,015
6
votes
2 answers

Elliptic curve with same number of points over two different fields

Following a discussion with a number theory professor, we arrived at the following question : Can we find an elliptic curve (short form : $y^2=x^3+ax+b$) with an identical number of points on two different finite fields: $GF(q)$ and $GF(q^r)$ with…
Thomas Lesgourgues
  • 4,685
6
votes
2 answers

Weierstrass equation long vs. normal form

So I am studying elliptic curves over finite fields and I am a little confused about something. In some texts I see a "long" Weierstrass equation and in some I see a "short" Weierstrass equation, what is the difference between the two? Are they…
Math Major
  • 2,234
5
votes
1 answer

division polynomials of elliptic curve as function on $\mathbb{C}$

I have a question about exercise 6.15 of Silverman's book AEC. Suppose that $E$ is a nonsingular elliptic curve over $\mathbb{C}$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Then we can define the division polynomials $\psi_n(x,y)$…
Nadori
  • 867
5
votes
0 answers

Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a generator point $G$, key $A$ (which is $aG$) and key…
user3081739
  • 185
5
votes
2 answers

Elliptic Curves - Identifying a Torsion Point, with some Rank thrown in

I'm a totally amateur mathematician who discovered elliptic curves this past summer and am quite fascinated by them. I am trying to learn more about them, but I am hampered by having studied Abstract Algebra 35 years and not remembering such things…
Regina Litman
  • 51
5
votes
2 answers

Real Period of an Elliptic Curve

Trying to work out what the real period of an elliptic curve is as seen in the Birch Swinnerton-Dyer conjecture. From what I've read, given an elliptic curve E over the rationals, one can associate to it a value $\displaystyle \Omega_{E} =…
CAB
  • 183
5
votes
0 answers

Elliptic curves: points with nonsingular reduction. Finiteness of a quotient.

Let $E$ be an elliptic curve defined over a complete local ring K in characteristic greater than 5 (one may assume that E is given by a Weierstrass equation of the form $y^2 = x^3 + Ax + B$). Denote by $\tilde{E}$ the reduction of E modulo the…
Marsan
  • 1,103
5
votes
1 answer

What is the rank of an elliptic curve?

Is it the amount of rational points on the curve(That aren't integers)? So if an elliptic curve has a rank of 1, does that mean it has only 1 rational point on the curve?(In both X and Y) I've been trying to search it up on google but it the…
Dean Yang
  • 387
1
2
3
…
26 27