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

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

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
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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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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 ...
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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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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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1answer
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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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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 ...
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1answer
59 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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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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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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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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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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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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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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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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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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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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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
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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]?