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

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2
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1answer
52 views

class number of pure cubic fields and elliptic curves

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will ...
0
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2answers
96 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 $C$ ...
0
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2answers
67 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 ...
3
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0answers
61 views

Elliptic curves as $\mathbb{C}^*/\mathbb{Z}$

I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where ...
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0answers
40 views

Computation of the 2-torsion group of an elliptic curve

I have some troubles solving the following problem: Let $E$ be the elliptic curve $E:y^2+2y=x^3+x+9$ over $\mathbb{F}_{16}$. Compute the 2-torsion group $E[2]$, i.e. find all the points of order $2$ ...
0
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1answer
76 views

Compute principal divisor for a rational function on a curve

During the lecture we defined the principal divisor of a rational function on a smooth curve as it follows: Consider the smooth curve $C\subseteq\mathbb{P}^2$. Take $g\in{K(C)^*}$. Then the principal ...
9
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2answers
90 views

order of an elliptic curve

I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact ...
3
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0answers
87 views

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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1answer
64 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 ...
1
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1answer
47 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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0answers
59 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 ...
7
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2answers
626 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the ...
1
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2answers
90 views

Elliptic Curve Crypto

I had just read a primer about ECC, I see how it can be complicated. Something I haven't been able to determine is what information does the client machine get to help decrypt the data? The whole ...
1
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3answers
74 views

The equation $b^2=a(a^2-1)$ has no rational solutions except obvious ones

I have problem with equation $b^2=a(a^2-1)$. How to show that except $(a,b)=(1, 0), (-1,0),(0,0)$, the equation hasn't any other rational solutions ? Editor's note. AFAICT the exercise is about ...
8
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1answer
147 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 ...
1
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1answer
30 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 ...
2
votes
2answers
175 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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2answers
57 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!
3
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1answer
93 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 ...
2
votes
1answer
54 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 ...
1
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1answer
34 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 $$ ...
3
votes
1answer
48 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 ...
0
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1answer
21 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 ...
0
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1answer
78 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) ...
3
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2answers
142 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 ...
5
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1answer
159 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 ...
0
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0answers
39 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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1answer
329 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
89 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 ...
2
votes
1answer
140 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 ...
0
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2answers
46 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
2
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0answers
69 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 ...
5
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2answers
318 views

Relation involving the conductor of an elliptic curve

Consider an elliptic curve $E: y^{2} = x^{3} + ax + b$. Then the quadratic twist by a squarefree $d$ is given by $E^{d} : dy^{2} = x^{3} + ax + b$. What is the relationship between the conductor of ...
0
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1answer
34 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.
1
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1answer
76 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 ...
2
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1answer
69 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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115 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
3
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1answer
63 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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0answers
71 views

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 ...
2
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1answer
71 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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0answers
75 views

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
297 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
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1answer
96 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. ...
4
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1answer
82 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 ...
4
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1answer
114 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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0answers
24 views

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 ...
1
vote
1answer
52 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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0answers
27 views

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 ...
4
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1answer
135 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
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0answers
47 views

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 $V\cap\mathbb{A}^n$, ...