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

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2
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1answer
59 views

Solving equations in $\mathbb{Z}_3$ with Hensel's Lemma

Further to the post here, I'm trying to find the $n \in \mathbb{Z}$ such that there is a solution to the equation $$ x^3 +3x+y^3+3y=n$$ in $\mathbb{Z}_3$. Now, I've been able to show that in the ...
3
votes
1answer
68 views

Real Lie groups and elliptic curves

Let $f:A\to A'$ be a morphism of elliptic curves over the real numbers $\mathbb R$. It canonically induces a morphism $f(\mathbb R): A(\mathbb R)\to A'(\mathbb R)$ between the sets of real points, ...
1
vote
1answer
65 views

hyperelliptic curve

Please help me to solve this question: Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. ...
3
votes
1answer
60 views

Elliptic Curve $E/\mathbb{Q}$ with $\Delta_E^{1/3}$ a root of defining cubic

Consider the elliptic curve $$E'\colon y^2 = g(x) = x^3 + \frac{1}{432} .$$ One can check that the discriminant of $E'$ is $-1/432$, that $E'$ has complex multiplication, and that $(-1/432)^{1/3} = ...
1
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1answer
79 views

Why so few complex multiplication

Let's take I lattice $\Lambda$ and $\alpha\in\Lambda$. Then we have $\alpha\Lambda\subseteq\Lambda$ so $z\mapsto \alpha z$ induces an isogenie $\mathbb{C}/\Lambda\to\mathbb{C}/\Lambda$ wich has no ...
1
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0answers
73 views

Group Operation of points on a Montgomery elliptic Curve (project coordinates)

I was trying to implement the double and addition formulas for elliptic points on a Montgomery elliptic curve. I came across this weird thing which should definitely not be happening. I took a point ...
4
votes
2answers
120 views

Can an elliptic curve have discrimant one?

Can the discriminant, $4a^3 +27b^2$ of an elliptic curve $$E: y^2=x^3+ax+b$$ be equal to 1. I believe that this should not be possible otherwise the curve would have good reduction at all primes $p$, ...
3
votes
1answer
73 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
3
votes
2answers
52 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
54 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
51 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
56 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 .
2
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0answers
42 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
50 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
votes
1answer
61 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 ...
3
votes
2answers
105 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
votes
1answer
108 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
32 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
votes
2answers
91 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
votes
2answers
48 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
vote
1answer
51 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 ...
6
votes
2answers
108 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 ...
6
votes
3answers
81 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 ...
1
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1answer
24 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 ...
8
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2answers
273 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 $1$, ...
2
votes
1answer
43 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, ...
2
votes
2answers
84 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 ...
1
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1answer
49 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 ...
5
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0answers
85 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
votes
1answer
52 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
83 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
votes
1answer
139 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 ...
2
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1answer
68 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 ...
1
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1answer
93 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 ...
1
vote
1answer
40 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 $k$ ...
3
votes
1answer
103 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
votes
1answer
107 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 ...
5
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0answers
117 views

Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
2
votes
2answers
179 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
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1answer
35 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 ...
4
votes
1answer
206 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 ...
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0answers
42 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
33 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 ...
1
vote
1answer
426 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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0answers
82 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 $ ...
11
votes
2answers
177 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
1
vote
1answer
35 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
votes
1answer
121 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
78 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. ...
1
vote
1answer
48 views

Point at Infinity of E.C. in Jacobian Coordinates

I am reading some notes about elliptic curves right now and the author mentions the alternative Jacobian projective coordinates, where one establishes the equivalence $(x,y,z)\sim (\lambda^2 x, ...