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

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Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
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1answer
31 views

On the rank of $y^2=x^3+a^2x^2-a^4x$

How can I prove that the rank of $y^2=x^3+a^2x^2-a^4x$ is zero where $a$ is rational and positive?
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1answer
50 views

Rational Number Form

I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state: Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers ...
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1answer
124 views

Rational points of order 2 on elliptic curves

I'm working on a question that's asking me to list all the $\mathbb Q$-rational points of order 2 and all the $\mathbb C$-rational points of order 2 for some elliptic curves. I've made the following ...
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0answers
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Reduction of isogenies at bad primes

Let $L$ be a number field and $E,E'$ two elliptic curves defined over $L$. Suppose $\varphi\colon E\to E'$ is an isogeny defined over $L$. Let $\mathfrak p$ be a prime of bad reduction for $E,E'$. ...
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1answer
47 views

Weierstrass normal form

How can I show that the Weierstrass normal form $u^3 + v^3 = \alpha$,with $x=12\alpha/(u+v)$ and $y=36\alpha (u-v)/(u+v)$, satisfy $y^2=x^3-432α^2$ ?
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1answer
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Rational Points on Elliptic Curves [closed]

Compute the group $C(F_p)$ for the curve $С : y^2=x^3 + x + 1 $ and the primes $p = 3, 7, 11$ and $13$.
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1answer
114 views

Are $10^{10}$-digit-numbers too big for Lenstra's elliptic curve method (ECM)?

I would like to search prime factors of the numbers $$10^{10^{10}}-113$$ and $$10^{10^{10}}+13$$ Both numbers have no prime factor below $10^9$. Are these numbers still too big for ECM ? I also ...
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1answer
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How to determine the group structure of $E(\mathbb{R})$ for an elliptic curve $E/\mathbb{R}$

Using Weierstrass' $\wp$ function it can be proved that the group of complex points on an elliptic curve $E /\mathbb{C}: y^2 = x^3 + ax + b$ satisfies $E(\mathbb{C}) \cong \mathbb{R}/\mathbb{Z} \oplus ...
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1answer
30 views

Complex Multiplication of $y^2=x^3+B$

I would like to find out what the complex endomorphism for the class of elliptic curves given by $$y^2=x^3+B$$ looks like. I know that for the class of elliptic curves $$y^2=x^3+Ax,$$ the complex ...
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1answer
54 views

Solutions of elliptic curve in finite field

If I take the following elliptic formula over a finite field of size $17$: $$y^2 = x^3 + 2x + 3$$ The solutions for $x = 2$ would be $7$ and $10$. Because $7^2=49$ and $49 \equiv 15 \bmod 17$ ...
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Using an elliptic curve to create pseudo random number

I recently started learning about encryption. I read about how elliptic curves can be used to create pseudo random numbers (and how the nsa might have abused this fact to create a backdoor in ...
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1answer
47 views

Formal group of an elliptic curve, from Silverman's the arithmetic of an elliptic curves

In the beginning of page 120, to establish the formal group law for an elliptic curve, the book adds 2 points $(z_1,w_1)$ and $(z_2,w_2)$, where $w_1 = w(z_1), w_2 = w(z_2)$ using the group law. It ...
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0answers
32 views

Inflection 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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1answer
71 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
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1answer
36 views

Size of automorphism group of element of $Y_0(5)$

The modular curve $Y_0(5)$ parametrizes elliptic curves $E$ with an isogeny of degree five. So an element of $Y_0(5)$ can be interpreted as $E \xrightarrow{\phi} E'$. Suppose we are working over a ...
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1answer
76 views

Weierstrass to general form

Can we go from short Weierstrass equation equation $y^2=x^3+Ax^2+Bx+C$ to general $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$?
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3answers
201 views

$ax^3+by^3+cz^3=0$ and Elliptic curves

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
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2answers
78 views

Elliptic Curve and Conjugation

If I consider an elliptic curve $C$ as a Riemann surface cut out in $\mathbb{C}P^2$ by a homogenous cubic, and if that cubic is defined over $\mathbb{R}$, then I think we have a conjugation map ...
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0answers
495 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
2
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1answer
101 views

Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
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0answers
44 views

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 ...
3
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1answer
66 views

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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0answers
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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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Are all elliptic curves from $w^3 = \text{cubic}(z)$ isomorphic?

I've been playing around with Riemann surfaces of cubics, and it seems to me that all surfaces obtained as coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a ...
2
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1answer
56 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 ...
2
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1answer
37 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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153 views

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
61 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
57 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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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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0answers
106 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
70 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
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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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1answer
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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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53 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
92 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
134 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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1answer
91 views

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 ...
2
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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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1answer
96 views

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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61 views

Infinite family of genus one non-elliptic curves over the rationals

How easy is it to write down genus one curves over $\mathbf Q$ without a rational point? Can we write down an infinite family?
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0answers
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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
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Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
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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 ...
3
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1answer
79 views

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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0answers
86 views

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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108 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 ...