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

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Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
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18 views

What is rational point on elliptic curve over Galois field

It is clear what is a rational point on elliptic curve, when the curve is defined over real numbers. But if it is defined over Galois field, what is a rational point? If necessary, supply an example, ...
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1answer
19 views

Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
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1answer
50 views

Seeking graphics of elliptic curves as surfaces

Is there any place to go on line for good graphics of how a complex elliptic curve sits as an affine curve in $\mathbb{C}^2$? The mathematics is well discussed in Drawing elliptic curve and Is the ...
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1answer
31 views

Surjectivity of p-adic representation

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} ...
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21 views

Non linear least square ellipse fitting

I am trying to find a Non linear leasts squares ellipse fit for a set of 100 data points data points $(x,y)$. Now i have found the values of $A,B,C,D,E,F$ according to the conical equation of the ...
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1answer
68 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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57 views

The Moduli Stack of Elliptic curves - What is it?

I have often heard the words "Moduli Stack of Elliptic Curves", but I have nowhere found a from-scratch definition of this object. I do understand the motivation: There are cusps in the moduli space ...
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1answer
19 views

Image of the p-Frobenius endomorphism under a mod p Galois representation.

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ a prime such that $E$ has ordinary redcution at p. Further, let $$\rho_{E,p}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm ...
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20 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
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2answers
41 views

Compendium of elliptic curves?

does anyone know where I can find a collection of elliptic curves and their integral solutions? EDIT: Removed additional useless info. Thanks!
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1answer
52 views

Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...
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30 views

Can we transform bijectionally each element in $C(ℚ)$?

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group. ...
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2answers
164 views

How to prove $~\sqrt{3}\sqrt{4a^{3}-1}~$ isn't an integer?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. If $~(c−b)=1~$ then $~c^3-b^3=3c^2-3c+1=a^3,~$ from Wolframalpha get: $$ c = \dfrac{3- \sqrt{3}\sqrt{4a^{3}-1}}{6} \\ c = ...
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1answer
42 views

Computing the trace of the image of Frobenius

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ and $q$ distinct primes and $e$ a positive integer. Fixing a basis for the $p^e$ torsion, we get a natural Galois representation $$\rho_{E,p^e}:{\rm ...
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1answer
28 views

Surjective $p$-adic representation implies trivial $p$-primary part.

Let $E/\mathbb{Q}$ be an elliptic curve. We know that by Serre in the non-CM case, for $p\geq5$, $$\rho_p:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(T_p(E))$$ is surjective iff $$ ...
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34 views

Curve avoiding semi-rational points

A rational point is a point in $\mathbb{R}^d$ all of whose $d$ coordinates are rational. Let me define a semi-rational point as one that has at least one rational coordinate (but whose other ...
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1answer
35 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
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1answer
18 views

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R ...
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1answer
41 views

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation).

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation). Assume $x_1 \neq x_2$. I create the straight line $y = m(x-x_1) ...
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2answers
55 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
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32 views

divisors eqivalence and Picard group

I have a Lemma which I understand until a certain point. It claims that if E is an elliptic curve and $P,Q \in E$, than $\exists R\in E:P+Q\sim R+\theta$, where $\theta$ is the point at infinity. To ...
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37 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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1answer
74 views

How to find primitive point on an elliptic curve?

Reading about Elliptic curve cryptography, i came across primitive point's or generator point's but found nothing on how to generate such points any help would be appriciated.
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1answer
57 views

Relating ramification index of a map of curves to degree of vanishing

I am little confused about explicitly computing ramification index and relating it to degree of vanishing a polynomial. In particular I have the following example (when trying to prove the genus ...
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1answer
93 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
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43 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
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1answer
29 views

How come $f(0) = 0$ in $\mathbb C/L$?

How come $f(0) = 0$ in $\mathbb C/L$? Does anyone know it? Your help will be appreciated. This is taken from the text "Rational Points on Elliptic Curves" by Tate and Silverman.
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1answer
112 views

Lenstra's Elliptic Curve Algorithm

I am currently trying to understand Lenstra's Elliptic Curve Algorithm for factoring integers. As a source I use "Rational Points on Elliptic Curves" by Joseph H. Silverman and John Tate. They ...
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1answer
48 views

Multiple points on an elliptic curve

I have given the following elliptic curve $E:F(x,y) = 0$: (Where $F(X,Y) := Y^2 + a_1XY + a_3Y - X^3 - a_2X^2 - a_4X - a_6$ with $a_1 = -1.5, a_2 = 3, a_3 = 1, a_4 = 0.5, a_6 = -1.5$.) The curve ...
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1answer
51 views

$p$-adic numbers and projective coordinates

Let $E/\mathbb{Q}_p$ be an elliptic curve and let $E^0(\mathbb{Q}_p)$ denote its nonsingular points. We accept that $E^0(\mathbb{Q}_p)$ is a subgroup of $E(\mathbb{Q}_p)$. Then let ...
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1answer
55 views

elliptic curves over $\mathbb{Q}_p$

Let $E: Y^2 = X^3 + AX + B$ be an elliptic curve over $\mathbb{Q}_p$, i.e. $A,B \in \mathbb{Q}_p$ and $4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if ...
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How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
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Cassel's book on Elliptic Curves

Let $E/\mathbb{Q}_p$ be an elliptic curve. Then for $n \geq 1$, let $E_n(\mathbb{Q}) = \left\{P \in E(\mathbb{Q}_p) : \dfrac{x(P)}{y(P)} \in p^n \mathbb{Z}_p\right\}$. According to Cassels in Lectures ...
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1answer
82 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
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1answer
43 views

Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
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Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
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1answer
85 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
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1answer
35 views

Find coordinate $y$ of an elliptic curve point

If I have an elliptic curve over a finite filed $F_p$ ($p$ is prime) defined as $$ y^2 \equiv x^3 + ax + b\pmod p,$$ such that $4a^2 + 27b^2 \neq 0$ and suppose I have only given the coordinate $x$, ...
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moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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Function field of a projective variety

I am reading Silverman's "The Arithmetic of Elliptic Curves". On page 10 he defines the function field of a projective variety $V$ over a field $K$ to be the function field of $V\cap\mathbb{A}^n$, ...
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1answer
39 views

Lattices and Elliptic curves and number fields

Let $K$ be a number field with ring of integers $O_K$. If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$. If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$. If $K$ ...
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59 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
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26 views

does every elliptic curve E/S have infinitely many sections after passing to an etale extension of S?

Let E/S be an elliptic curve, where S is any scheme. Must there exist a scheme $S'$, etale and surjective over $S$, such that the pullback $E' := E\times_S S'$ has infinitely (or even > 1) many ...
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119 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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1answer
42 views

Reduction map on torsion of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both ...
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14 views

What is an embedding degree of elliptic curve?

I am dealing with MOV algorithm to transform ECDLP to DLP in $GF(p^k)$, but at the first step I have to determine embedding degree k. I have read the definitions of embedding degree, but still I am ...
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1answer
31 views

Silverman AEC Corollary 6.4

Quick question about Chapter 3 Corollary 6.4 [p. 86] in Silverman's Arithmetic of Elliptic Curves. I feel like I'm misreading it and would like clarification. He claims that for an elliptic curve E ...
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1answer
55 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same ...
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1answer
61 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...