For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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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 $E$.
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1answer
102 views

Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
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1answer
76 views

Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
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1answer
151 views

Characterization of the $m$-torsion points of an elliptic curve.

Let $(E,\mathcal{O})$ be the elliptic curve of equation $$ f=Y^{2}+a_{1}XY+a_{3}Y-X^{3}-a_{2}X^{2}-a_{4}X-a_{6}, $$ $\alpha:K(E)\rightarrow K(E)$ the derivation such that $$ \alpha(X)=\frac{\...
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Elliptic curve characteristics 2 and 3

How can you show that if the characteristic of an elliptic curve $y^2 = x^3 + ax + b$ is 2 or 3 the equation fails? For characteristic 2 I know the equation must be written as $y^2 + ay = x^3 + bx^2 + ...
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1answer
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Is the sheaf of differentials on an elliptic curve over $R$ with a Weierstrass equation free?

Let $R$ be an integral domain and $E\stackrel{f}{\rightarrow}\text{Spec }R$ be an elliptic curve given by $$E := \text{Proj }R[x,y,z]/(y^2z + a_1xyz + a_3yz^2 = x^3 + a_2x^2z + a_4xz^2 + a_6z^3)$$ ...
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An implement of Constructing elliptic curves of prescribed order

In the Reinier Bröker's Phd thesis——Constructing elliptic curves of prescribed order(2006), he present a effective way to generate a elliptic curve with a given order N. And the heuristic run time of ...
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1answer
320 views

Understanding proof by algebraic geometry, Fermat's last theorem for polynomials when $n = 3$.

This is a followup to my question here. See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in \...
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0answers
38 views

Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia (https://en.m....
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1answer
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Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
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70 views

Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
2
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1answer
65 views

The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested ...
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0answers
46 views

$E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...
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72 views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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Canonical sheaf of bielliptic surface

This is an example of bielliptic surface on page 84 from Beauville's book "Complex algebraic surfaces". Let $\rho^3=1$, $\rho\neq1$ and $F_\rho=\mathbb{C}/(\mathbb{Z}+\rho\mathbb{Z})$, $G=\mathbb{Z}/...
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1answer
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How can I find $E(\mathbb F_{17})$ for the elliptic curve $E:$ $y^2=x^3+c$ where $c$ is any element in $\mathbb F_{17}^*$?

This was left as an exercise in a seminar in my college. I tried to figure it out myself, but haven't been able to make any progress thus far. I don't think it should need any non-trivial result (or ...
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1answer
170 views

Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) \otimes_\...
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2answers
97 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = ...
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0answers
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Understanding the group structure of quotient group derived from elliptic curve group

I am working through some content in L.C. Washington's Elliptic Curves, Number Theory, and Cryptography and I am unsure about what the group structure of a certain group looks like. Some background: ...
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Is the coordinate ring of an elliptic curve principal?

Let $K$ a field, $E$ an elliptic curve. I would like to know if the coordinate ring of $K[E]=K[X,Y]/(E)$ is principal. I think the answer is no. I tried to prove that the ideal $J=\langle y,x^2\rangle$...
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1answer
106 views

find the structure of an elliptic curve over a finite field

For the elliptic curves E1,E2,E3, and E4 defined below, determine the structure of the groups Ek(F13) by using the information given below together with a minimal amount of extra (hand) ...
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1answer
75 views

An elliptic curve has a point of order $n$ iff $E[n]\cong\mathbb{Z}/n\times\mu_n$?

Let $E$ be an elliptic curve over some field $k$, and let $n$ be coprime to the characteristic of $k$. Then I claim that $E(k)$ has a point of order $n$ if and only if $E[n]\cong\mathbb{Z}/n\times\...
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1answer
36 views

Real elliptic curves

Is it true that the full automorphism group of a real elliptic curve is $T\rtimes\mathbb{Z}/2\mathbb{Z}$ where $T$ is either $SO_2(\mathbb{R})$ or $SO_2(\mathbb{R})\times\mathbb{Z}/2\mathbb{Z}$? If '...
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24 views

Moduli Space of elliptic fibration

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration. I know how it is done for the generic Weierstrass ...
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59 views

Cuspform, elliptic curves and character sums

I was trying to read through some note by Kowalski (see https://people.math.ethz.ch/~kowalski/ik-ant-exp-sums.pdf). I was interested in trying to understand the following. The author states on page ...
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1answer
45 views

Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
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1answer
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Unramified cocycles and the Selmer group of an ellptic curve

In Silverman's book on elliptic curves, he gives a procedure to compute the Selmer group of elliptic curve $E$ relative to an isogeny $\phi:E\to E'$. I am confused about one step in the discussion. ...
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1answer
29 views

Better parametrization for computing group law of nodal cubic?

In undergrad, I remember computing the group law of the nodal cubic $y^2= x^3 + x^2$ using a particularly slick parametrization. The usual parametrization of the nodal cubic is $(t^2-1, t^3-t)$, and ...
3
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1answer
122 views

Finding two non-congruent right-angle triangles

The map $g: B \to A, \ (x,y) \mapsto \left(\dfrac {x^2 - 25} y, \dfrac {10x} y, \dfrac {x^2 + 25} y \right)$ is a bijection where $A = \{ (a,b,c) \in \Bbb Q ^3 : a^2 + b^2 = c^2, \ ab = 10 \}$ and $B =...
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1answer
89 views

Is there something similar to $\mathbb{R}^2$ for elliptic curve point representation?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$. Consider a coordinate system in $\mathbb{R}^2$. Every point is ...
3
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1answer
48 views

$J$ invariant of elliptic over a number field

Suppose $E$ and $E’$ are elliptic curves over a number field $K$ which are Galois conjugate over $\mathbb Q$. So $\operatorname{End}_C(E)$ and $\operatorname{End}_C(E’)$ are isomorphic. Suppose $\...
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26 views

What is the rationale behind change of variables in elliptic curves?

Say we have an elliptic curve in its most general form: $Ax^3 + Bx^2 y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + Hx + Iy + J = 0$ Many websites say that "through appropriate change in variables," we can ...
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0answers
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Relationship between discriminants and smoothness of curves

My understanding of the use of the discriminant in elliptic curve theory is to test whether an elliptic curve in Weierstrass normal form over a field not of characteristic either 2 or 3, $y^{2} = x^{3}...
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36 views

Direct sum of two points on an elliptic curve

Given $E:y^{2} = x^{3}+9x$ over $\mathbb{Z}_{71}$, and $A = (0,0), \: B = (1,9)$, I'm asked to find $C=A\oplus B$. I just don't know how the direct sum of two points on an elliptic curve is defined, ...
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207 views

The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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1answer
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Proving some facts about the EC $y^2 = x^3 + ax + b$ [closed]

A solution to this question would be much appreciated! If $E/F$ is the EC defined by $y^2 = x^3 + ax + b$ then prove the following: If $P = (x, y)$ is element of $E(F)$ with order 3 then $x$ is a ...
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Weil pairing of curve of genus 2

We know there is Weil pairing for elliptic curve satisfying several nice properties. So do we have Weil pairing for other curves also satisfying the nice property? Especially genus 2 curve?
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2answers
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Quotient of invariant differentials is constant

In the proof of Proposition 2.1.1 in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, he makes a comment about quotients of invariant differentials being constant, because their ...
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Showing $N_E/F_p^r = p^r + 1$ for a given elliptic curve [closed]

Have been studying elliptic curves, and am syuck on this problem. A detalied explanation would be much appreciated! a. Let $E/F_p^r$ be the elliptic curve $y^2 = x^3 − x$. Prove that if $p ≡ 3 (mod 4)$...
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Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P ∈ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
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125 views

Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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1answer
61 views

2-torsion points in a curve with genus 2

Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in ...
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1answer
118 views

Finding order of a point on eliptic curve

Just started studying eliptic curves and am having trouble with this question. An explanation/solution would be much appreciated. Find the order of the point X on the elliptic curve $E/Q$ for the ...
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1answer
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Is the image of $\overline{\rho_{E,p}}$ in $PGL_2$ always isomorphic to $A_5$ if $p$ does not divide the order of the image of $\rho_{E,p}$?

I have the following setting: Given an elliptic curve $E$ over $\mathbf{Q}$ and $p>5$ a prime of good ordinary reduction. Let ${G}_{k}=\text{Gal}(K(E_{p^k})/K)$ with representation $\rho_{E,p^k}:G\...
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Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
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1answer
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How to construct isogenies between elliptic curves over finite fields for some simple cases?

From the theorem of Tate, it is known that the two elliptic curves over the same field $\mathbb{F}_p$ are isogenous iff they have the same number of points. For $p\equiv 3\mod 4$, the curve $E_1(\...
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20 views

Hensel lemma and elliptic curves [closed]

What is the cardinality of this homomorphism $E(\mathbb{Z}/p^r\mathbb{Z} ) \rightarrow E( \mathbb{Z}/p\mathbb{Z} )$ where $E(\mathbb{Z}/p^r\mathbb{Z} )$, $E( \mathbb{Z}/p\mathbb{Z} )$...
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25 views

Is the connection between elliptic curve and lattice unique?

If I remember correctly, elliptic curve (over C) is isomorphic to a complex lattice, and they are connected by some technical stuff(Eisenstein series, j-invariant,...) But the whole process seems so ...
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1answer
53 views

Is there another methods for counting points on the curve $x^3 + y^3 =1$ over finite fields?

For the circle $(C): x^2 + y^2=1$ over finite field, we can use simple method to count the number of points. The case $p\equiv 1\mod 4$ is not difficult to find, because $-1$ is a square on $F_p$. ...