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

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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 ...
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1answer
74 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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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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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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98 views

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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99 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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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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2answers
153 views

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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55 views

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
39 views

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
86 views

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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259 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
34 views

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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105 views

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
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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126 views

Challenge from Fermat

Fermat challenged Frenicle with finding a pythagorean triple (a,b,c) where $(a-b)^2-2b^2$ is itself a square. By making the substitution $a=m^2-n^2$, $b=2mn$, and $c=m^2+n^2$ into $(a-b)^2-2b^2=d^2$ ...
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1answer
47 views

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
82 views

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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Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ ...
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Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
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135 views

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 ...
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181 views

Uncertain about Uniformizing Elements of Elliptic Curves.

I am following a subject on Elliptic Curves and have come accross the notion of a uniformizer. Wikipedia tells me that an element is a uniformizer of a Discrete Valuation Ring, if it generates the ...
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1answer
111 views

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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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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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
157 views

Why must the order of basepoint of elliptic curve be prime?

Let $E$ be an elliptic curve defined over a finite field $F(q)$. Let $G\in E(F(q))$ be a point of order $n$, where $n$ is a prime number and $n>2^{160}$. The elliptic curve discrete logarithm ...
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Complexes Torus and $\mathrm{PSL}_2(\mathbb{Z})$

I want to prove that if $\omega_1\equiv\omega_2$ modulo $\mathrm{PSL}_2(\mathbb{Z})$ then $X(\omega_1)\simeq X(\omega_2)$ where $X(\omega)=\mathbb{C}/(\mathbb{Z}+\omega\mathbb{Z})$. I see that if ...
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1answer
80 views

Prove that $y^2 = x(x-1)(x- \lambda)$ is irreducible for all $\lambda \in k$

I wish to prove that $y^2 = x(x-1)(x- \lambda)$ is irreducible for all $\lambda \in k$. It seems like this follows from the fact that $x(x-1)(x- \lambda)$ cannot be written as the square of any ...
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Supersingular elliptic curves- Invariant differential exact proof question

I'm writing a minor thesis about different criteria of supersingularity and I wanted to show the following from Husemöller's Elliptic Curves [Prop. 13.3.8]: An elliptic curve $E$ in characteristic ...
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How one can determine the rapidly convergente series?

The motivation to this question can be found in http://wstein.org/books/bsd/bsd.pdf In page 9 the author claimed that: 1.4.1. Approximating the Rank. Fix an elliptic curve $E$ over $Q$. The usual ...
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2answers
92 views

How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
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50 views

Hash functions that produce a point on an elliptic curve.

I see in some cryptographic papers, that the authers of those papers utilize of a hash function such that that hash function converts an integer value or octet-string value as input, to a point on an ...
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Prove that the only negative real zeroes are at the integers

Let $$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse-Weil $L$-function of ...
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825 views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
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64 views

This proof is completely unclear for me

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. I find the following proof in an old Russian book: I want to ...
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190 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
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197 views

Subtraction two points on elliptic curve.

Suppose Q, T and S are three points on an elliptic curve, such that Q+T = S. With knowing Q and S, can we compute T? In other word whether exists subtraction operation on elliptic curve, or not?
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Addition, Subtraction points on an elliptic curve with integer values

I see in some papers that their authers claimed that with operation such as XOR, Addition and subtraction, hide an integer value by a point on an elliptic curve. to clarify, suppose Q is a point in ...
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1answer
52 views

How to relate the valuation of x/y (For a minimal Weierstrass equation)

I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as: "Let $K/\mathbb{Q}$ be a number field and $E/K$ an ...
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1answer
59 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
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1answer
240 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
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Addition a point on an elliptic curve with an integer value

Suppose $Q$ is a point in an elliptic curve such that $Q=dP$ and $d$ is an integer value, and $P$ is base point of that elliptic curve. Note $Q = dP$ means that $P+\cdots+P$ for $d$ times** and since ...
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Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
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61 views

Can one explain to me this Theorem

Can one explain to me Theorem 2. (page 3) in this link: http://www.math.leidenuniv.nl/~evertse/siksek-modular.pdf I am but confused about the nature the bijection defined in that result.
4
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1answer
294 views

Square root in Characteristic 2 Field

Let $K$ be a field of characteristic 2. For each $a\in K$, can we always find some $x$ such that $x^2=a$? I came upon this question while reading "Arithmetic of Elliptic Curves". The original ...
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203 views

Making an elliptic curve out of a cubic polynomial made a cube, or $ax^3+bx^2+cx+d = y^3$

What is the transformation such that a general cubic polynomial to be made a cube, $$ax^3+bx^2+cx+d = y^3\tag{1}$$ can be transformed to Weierstrass form, $$x^3+Ax+B = t^2\tag{2}$$ (The special ...
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1answer
79 views

Diffie-Hellman key exchange for three user.

Assume that there are three users that have their own secret key $d_i$ and corresponding public key $Q_i = d_i G$ such that $Q_i$ is a point in an elliptic curve. Now I'm looking for a solution to ...