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

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Infection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is ...
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Embedding of elliptic curves into $\mathbb{P}^2$ by arbitrary line bundle of degree $3$

Let $E$ be a complex elliptic curve, with distinguished point $x_0 \in E$. Any divisor of degree three is equivalent to the divisor $D=x+2x_0$. If $x=x_0$, it is well known that $L(D)$ has an explicit ...
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Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in ...
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1answer
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Is it possible to compute order of a point over Elliptic curve?

In the elliptic Curve cryptography, it is said that the order of base point should be a prime number, and order of a point $P$ is defined as $k$, where $kP = \mathcal{O}$. And to compute the order we ...
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1answer
176 views

Help in finding curve equation.

What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the ...
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1answer
55 views

Projective coordinates for elliptic curves

If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)? I'm ...
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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 ...
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Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
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1answer
35 views

Uniqueness of points in Elliptic Curve addition

When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!
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1answer
58 views

Calculating Non-Singular Map of Elliptic Curve

I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity ...
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1answer
53 views

Prove that the Frobenius map is a homomorphism

I want to prove that the Frobenius map $\phi$ is a homomorphism from the group of points on an elliptic curve $E(F_{2^k})$ to itself (endomorphism). It is trivial to check that if a point $P \in E$ ...
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Probability of an ECM factor

Suppose I have a composite number $N$ divisible by some prime $p\le x.$ What is the probability that one iteration of ECM finds $p$, given parameters B1 and B2? Usually people look for factors in ...
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102 views

Representing Points of Jacobian in Magma

Although I understand the Mumford representation of points on the Jacobian (of a genus 2 hyperelliptic curve), I don't understand how Magma represents such points. I would guess the confusion arises ...
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1answer
62 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
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1answer
75 views

Counting the number of elliptic curves with certain discriminant/conductor

I'm looking for some references regarding the above topic. To be more specific, references that address questions such as Given $D > 0$, how many elliptic curves over $\mathbb{Q}$ are there with ...
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1answer
28 views

ellipse chord length along its axis.

how to determine the position in an ellipse, where the chord length is equal to its minor axis and perpendicular to the major axis? Is there any equation for it?
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47 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
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What is so special about frobenious endomorphisms in elliptic curves?

What is so special of Frobenious endomorphisms in elliptic curves?Why we use it for? Does it have any severe implication?
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1answer
90 views

The evaluation map for a skyscraper sheaf on an elliptic curve

Let $E$ be an elliptic curve over a field, $z \in E$ is a point, and $d \geq 1$. I consider a skyscraper sheaf $\mathcal{O}_z/m_z^d$, the evaluation map $$ \operatorname{Hom}(\mathcal{O}, ...
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2answers
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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 ...
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1answer
148 views

Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in ...
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1answer
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complex multiplication in elliptic curves

The following question is in my homework: How many complex elliptic curves (up to isomorphism) have complex multiplication by the ring $\mathbb{Z}[\frac{1+\sqrt{D}}{2}]$ of discriminant $D=-71$ and ...
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1answer
127 views

Group structure of an elliptic curve

Let $E$ be an elliptic curve over field $\mathbb{Z}/p\mathbb{Z}$. The curve group $E(\mathbb{Z}/p\mathbb{Z})$ is always a) cyclic or b) direct product of two cyclic groups. First question: How do I ...
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The group of $\mathbb{K}$--rational points for isomorphic elliptic curves

The Springer text by Tom Apostol on Dirichlet series and modular forms, which I have, defines modular functions and modular forms on page 34 and on page 114 respectively, not to mention the Springer ...
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1answer
95 views

Order of a subgroup on elliptic curve over a finite field

May i ask you for a little help for a problem about elliptic curves? Here's the problem: Given an elliptic curve $E$ over the finite field $\mathbb{F}_{101}$. We know that there is a point of ...
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Automorphisms of elliptic curves with a point of order N

Probably a stupid question, but I'm trying to figure out the following: Suppose we have an element of $X_1(N)$, i.e. an elliptic curve $E$ with a point $p$ of order $N$. If we have another such curve ...
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1answer
67 views

Order of subgroup on elliptic curve over $Z_p$

I should determine the order of subgroup on elliptic curve over $\mathbb{Z}_p$ where $p$ is prime, and point $X$ is generator of some subgroup. While generating the subgroup by points addition I found ...
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Frobenius Endomorphism

I had a lecture last week which dealt with the Frobenius Endomorphism on elliptic curves. The lecturer showed an example at the end of the lecture, when almost out of time and I don't quite understand ...
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104 views

Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be ...
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1answer
53 views

Find the bound for [K(E[p]):K]

Let E be an elliptic curve over a field K of characteristic p > 0, we know that E[p] has order 1 or p, how to bound [K(E[p]):K]?
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Complex Multiplication/ Elliptic Curves Question

I'm working on the following problem from Silverman's advanced topic in the arithmetic of elliptic curves: Let $E$ be an elliptic curve defined over a number field $L$ with complex multiplication by ...
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1answer
57 views

Point multiplication in elliptic curve

Suppose $a$ is an integer and $Q$ is a point on an elliptic curve and $(x,y)$ are $x$ and $y$ coordinates of this point. My question is: Whether $a\cdot Q$ is equal to $(ax, ay)$?
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Transform Weierstrass equation into cubic

How can I transform an elliptic curve over the real numbers in Weierstrass form $y^2=x^3+ax+b$ into a cubic of the form $y^2=x(x-c)(x-d)$?
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1answer
100 views

Elliptic Curve: Multiplying points over a finite field

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ where $q$ is prime. Let $P$ be a point on $E$. Consider the point $Q=(q+1)P=P+\cdots+P$, which is $P$ added to itself $q+1$ times. Due ...
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Does there exist an elliptic curve $E$ such that $\#E(\Bbb{F}_{q^2})=(q+1)^2$ for all prime powers $q$?

The following (paraphrased) question is a homework exercise for a course on elliptic curves: Let $p\not\equiv1\pmod{12}$ be a prime number and let $q=p^k$. Show that there exists an elliptic curve ...
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Isomorphism of Elliptic Curves:

In Stinson's Cryptography Theory and Practice, a theorem is given without proof: Theorem 6.1 Let $E$ be an elliptic curve defined over $Z_p$, where $p$ is prime and $p > 3$. Then there exist ...
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1answer
62 views

Counting elements of $y^2 - y = x^3$ in finite fields

The problem I have to solve is the following: Let $p$ be a prime number with $p \equiv 2$ mod $3$. Let $E$ be the elliptic curve given by $y^2 - y = x^3$. Show that $\#E(\mathbb{F}_p) = p+1$ and ...
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Amount of points on an elliptic curve over $F_q$

Assume I have these two elliptic curves: \begin{align*} E:Y^2&=X^3+b_2X^2+b_4X+b_6\\ E':Y^2&=X^3+gb_2X^2+g^2b_4X+g^3b_6, \end{align*} over $\mathbb{F}_q$, where $g$ is not a square in ...
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1answer
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Quadratic twist of an elliptic curve

I found this page: http://en.wikipedia.org/wiki/Twists_of_curves#Quadratic_twist which tells me $dy^2=x^3+a_2x^2+a_4x+a_6$ is equivalent to $y^2=x^3+da_2x^2+d^2a_4x+d^3a_6$. Why is this equivalent ...
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1answer
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Number of points on $Y^2 = X^3 + A$ over $\mathbb{F}_p$

Let $p\equiv 2\pmod{3}$ be prime and let $A\in\mathbb{F}^{∗}_p$ . Show that the number of points (including the point at infinity) on the curve $Y^2 = X^ 3 + A$ over $\mathbb{F}_ p$ is exactly $p + 1$ ...
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1answer
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Does this equation have a rational point? (Elliptic curve?)

Can someone check pls if, $$852 + 3017 x - 1104 x^2 + 2009 x^3 - 3362 x^4=y^2$$ has a rational point? (This arose in an equal sums of like powers problem.) P.S. I've checked $x=p/q$ for ...
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262 views

Why 1728 in $j$-invariant?

The $j$-invariant for elliptic curves has a $1728$ in it. According to Hartshorne, this is supposedly for characteristic-$2$ and $3$ reasons, despite appearances to the contrary. Indeed, it is ...
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1answer
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isogenies between tori

Let Hom$(\mathbb{C}/\Lambda_1,\mathbb{C}/\Lambda_2)$ be the set of isogenies between $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$, where $\Lambda_1,\Lambda_2$ are lattices. I am asked to prove ...
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State of the art in arithmetic moduli of elliptic curves?

In trying to get into the topic of moduli spaces of elliptic curves, the following question arises: What is the state of the art in the topic right now? Deligne and Rapoport describes how the ...
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1answer
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summing points in elliptic curve

Does $P_1+P_2+P_3=2Q$ where $\{P_i\}_{i=1}^3,Q\in E(\Bbb F_p)$ for an ellliptic curve $E/\Bbb F_p$ mean $Q\in\{P_i\}_{i=1}^3$? I think I could just ask does $P_1+P_2=2Q$ where $\{P_i\}_{i=1}^2,Q\in ...
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1answer
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Genus of Edwards curve

Let us work over a field $\Bbbk$ of characteristic not equal to two. Let $d\in\Bbbk\setminus\{0,1\}$. It is said in the wikipedia article about Edwards curves that the plane quartic defined by the ...
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1answer
90 views

Finite order points on elliptic curves

Let $E = V_+(F(u,v,w)) \subset \mathbb{P}^2_k$ be an elliptic curve. Let $o = (0,1,0)$ be the origin and $x \in E(k)$ a rational point. Let us suppose there is a curve $C \subset \mathbb{P}^2_k$ such ...
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Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...
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1answer
72 views

Weierstrass form for some equation

How to find a birational transformation that turns the equation $3(y^2-1)=2x^2(x^2-1)$ into Weierstrass form? Thanks!
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2answers
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Question about uniformizers of Elliptic Curves

Let $k$ be a field with $Char(k)\neq 2,3$ and $E: y^2=x^3+Ax+B$ an elliptic curve over $k$ , where $4A^3+27B^2\neq 0$ and let $P=(\alpha,\beta)$ be a point defined over $k$. Show that if $\beta\neq ...