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

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3
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24 views

Faster scalar multiplication over an elliptic curve by hand?

For an elliptic curve $y^2=x^3+ax+b$, I have $a=1, b=1, G=(3,10)$ private key of User $B$ as $4$. To calculate his public key, I have the formula: $Pb=nb \times G = 4(3,10)$. This makes my ...
1
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1answer
38 views

Dividing elliptic curve point on integer

Can we solve the equation $nP = Q$, where $P$, $Q$ is a rational poins on elliptic curve ($P$ is unknown), and $n$ is integer?
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1answer
42 views

Group on elliptic curve points

Let's we have an elliptic curve (EC). Is it possible to construct group $G$ acting on the points of an EC with this property: if $P$ is a rational point on EC then $G(P)$ is also is a rational point? ...
0
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0answers
49 views

How I can determine the number of integral solutions of the equation of the elliptic curve?

Is there a general law to determine how many integral solutions of the Equation of the form (elliptic curve): $y²=x^{3}+ax+b,\ \ \ a,b \in \Bbb R$. Thank you for any help .
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0answers
22 views

Computing the order of a divisor in the Jacobian of a hyperelliptic curve.

Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over ...
1
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1answer
27 views

Number of points on an elliptic curve and it's twist over $\mathbb{F}_p$.

I have another probably very trivial question about elliptic curves. This wikipedia article gives the following formula $|E|+|E^d|=2p+2$ where $E$ is an elliptic curve over $\mathbb{F}_p$ and $E^d$ is ...
2
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1answer
41 views

Pole Order of Weierstrass Coordinates

I'm trying to understand a proof in Silverman's The Arithmetic of Elliptic Curves. Background: For an elliptic curve $E$, $x, y \in K(E)$ such that $\phi = [x, y, 1]: E \rightarrow C$ is a basepoint ...
5
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0answers
142 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
3
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2answers
80 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
3
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1answer
93 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
0
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1answer
29 views

How to calculate an elliptic curve

I need to find an elliptic curve in $F_{19}$ that has $|E(F_{19})|=18$. I am really stuck here. Can anyone help?
2
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2answers
47 views

Real points of order 3 on an elliptic curve.

This comes from Silverman's Rational Points on Elliptic Curves: Consider the elliptic curve (non singular) $y^2=x^3+ax^2+bx+c=f(x)$ after some computations we can see that points of order 3 in this ...
2
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2answers
46 views

Elliptic curve with prescribed lattice

It's well known that there is a connection between elliptic curves and lattices. To establish such a connection one needs to use Eisenstein series. How one can one write down the explicit equation of ...
1
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1answer
39 views

Finding a rational point on $\mathscr{E} : y^2=x(x^2-25)$ to show $ \text{rank}(\mathscr{E})=1$

I'm trying to show that the rank of the following elliptic curve $$ \mathscr{E}: y^2=x(x^2-25)$$ is 1. Since it has a rational 2-torsion point at $(0,0)$, by considering the dual curve I've been ...
8
votes
2answers
186 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
6
votes
2answers
268 views

Other ways to compute the torsion subgroup of elliptic curves

Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this ...
5
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2answers
79 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
8
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2answers
211 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
6
votes
3answers
75 views

Semigroup law on points on the curve $f(x) = \frac{1}{x}$

Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$. Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve. Draw a line through them Find where this point ...
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1answer
15 views

Converting to homogenuous coordinates

Let's assume elliptic curve $E$ over $\mathbb{R}$: $y^2 = x^3 + x + 1$ How to convert this equation to homogeneous coordinates? My notes say it's $zy^2=x^3+xz^2+z^3$. Unfortunately, I have no idea ...
3
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1answer
69 views

When is a modular curve defined over Q?

Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) ...
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1answer
42 views

Interpretation of a short exact sequence from elliptic curves in terms of torsors

Consider some elliptic curve $E$ over a number field $k$. Then for any prime $p$ there is a short exact sequence $$ 0 \to E(k)/pE(k) \to H^1(k,E[p]) \to H^1(k,E)[p] \to 0. $$ Now, $H^1$ has an ...
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0answers
65 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
2
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2answers
59 views

Types of elliptic curves

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 ...
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1answer
23 views

Proof of Theorem III, 6.2 in Silvermans Arithmetic of Elliptic curves

In the proof of part c), I cannot make sense of the sentence "Then another way of saying that $\phi:E_1\to E_2$ is an isogeny is to note that $\phi(x_1,y_1)\in E_2(K(x_1,y_1))$." (In this situation, ...
7
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1answer
125 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 ...
2
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1answer
45 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
2
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1answer
80 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
2
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1answer
46 views

A question about endomorphism rings of elliptic curves

This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it. Assume we have an elliptic curve $E$ over ...
2
votes
1answer
50 views

Question about characteristic polynomial of the Frobenius endomorphism on elliptic curves.

I have another possibly trivial question about elliptic curves. A lot of papers I've seen state that the characteristic polynomial of the Frobenius endomorphism of an elliptic curve over a finite ...
3
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3answers
424 views

How do you determine if an elliptic curve over a finite field is cyclic?

I know the group order and the points of the elliptic curve $y^2 = x^3 + Ax + B$, but I am confused on how to determine if they from a cyclic group The curve $y^2 = x^3 + 2x +2$ in $\Bbb F_{11}$ ...
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1answer
74 views

Elliptic curves and Weierstrass $\wp$ function - an example

Let $E: y^2 = 4x^3 - b$ an affine equation of an elliptic curve in $\mathbb{P}^2_{\mathbb{C}}$. Let $b$ be chosen such that the map $f: \mathbb{C} \rightarrow \mathbb{P}^2$ given by $z \mapsto ...
4
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1answer
191 views

Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
4
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1answer
117 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
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0answers
56 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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1answer
36 views

Benefit from local coordinates

I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows: Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ ...
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2answers
72 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m ...
3
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1answer
37 views

Question on why short Weierstrass can't be used for curves with char=2

An elliptic curve given by $E: y^2=x^3+ax+b$ with $a,b \in K$ and $Δ(E)=-16(4a^3+27b^2) \neq 0$ is adequate for elliptic curves with $char\neq2,3$ Because of the factor -16 in the definition of ...
2
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1answer
50 views

Topics in elliptic curves over finite fields

First of all, sorry if I didn't put this question in the correct category. This a paper aimed for undergraduate math majors. So I am writing a general paper explaining about elliptic curves over ...
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0answers
79 views
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0answers
51 views

Genus of the product of two elliptic curves

In trying to understand the trichotomy of the genus of algebraic curves, I first consider the following two elliptic curves (over $\mathbb{Q}$), well-known to be of rank $2$, $ y^2 = x^3+17$ and $ ...
2
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2answers
42 views

Weierstrass equation long vs. normal form

So I am studying elliptic curves over finite fields and I am a little confused about something. In some texts I see a "long" Weierstrass equation and in some I see a "short" Weierstrass equation, what ...
2
votes
1answer
28 views

A clarification of addition on elliptic curves over the complex numbers

I am trying to prove that the order of the two points $P_{\pm}=(0,\pm\sqrt{-g_3})$ is three on the elliptic curve $y^2=4x^3-g_3$, for $g_3 \not= 0$, defined over $\mathbb{P}^2_{\mathbb{C}}$. Here's an ...
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0answers
40 views

$x^3+y^3+z^3 = 0$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega \not= 1\}$

I am rather stuck trying to prove that $x^3+y^3+z^3 = 0$ in $\mathbb{P}^2_{\mathbb{C}}$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega ...
2
votes
1answer
25 views

Elliptic curve notation

This might be kind of a silly question about notation. I know: $E$: an elliptic curve $\mathbb{F_q}$: finite field But I recently ran across the notation $E/\mathbb{F_q}$ for the first time, so ...
11
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2answers
161 views
1
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1answer
78 views

Hasse's Theorem for Elliptic Curves over Finite Fields + proof clarification

I need a little help understanding Hasse's theorem for elliptic curves over finite fields, as well as the proof of this theorem. (Sorry about my editing) Hasse’s Theorem: Let $E$ be an elliptic ...
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1answer
19 views

Sato-tate conjecture for elliptic curves over finite fields

I am doing a research project about elliptic curves over finite fields and I am across the Sato-tate conjecture, but I am having some difficulty understanding it. What I (think) I took away from the ...
2
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1answer
43 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
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2answers
42 views

Chinese Remainder theorem on Elliptic Curve group

I read somewhere (Blake, Seroussi, Smart: Elliptic Curves in Cryptography, p.160) that one can use the Chinese Remainder theorem to split $E(\mathbb{Z}/N\mathbb{Z})$, where $N$ is a composite number. ...