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

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2
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2answers
63 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 $p \equiv 3\mod 4$, ...
0
votes
0answers
26 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
1answer
70 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
25 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
46 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
44 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
32 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 ...
-1
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0answers
49 views

how to find solution(all points ) of $y^2+xy=x^3+x^2+1$? [closed]

actually ,this is elliptic curve(koblitz curve) and I want to know all the points on this curve. so would you please tell me how to find points on this curve defined over finite field(F 2^4)??
8
votes
2answers
168 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
262 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
votes
2answers
67 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 ...
7
votes
2answers
161 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
54 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
vote
1answer
14 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
votes
1answer
60 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) ...
1
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1answer
35 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
votes
0answers
53 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} = ...
5
votes
1answer
100 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
2
votes
2answers
47 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
vote
1answer
17 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
votes
1answer
124 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
votes
1answer
43 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
votes
1answer
77 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
43 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 ...
3
votes
1answer
36 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
votes
3answers
385 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 $\Bbb F_{11}$ ...
1
vote
1answer
63 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
votes
1answer
187 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
votes
1answer
108 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 = ...
1
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0answers
53 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 ...
1
vote
1answer
33 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 $k$ ...
1
vote
2answers
66 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
votes
1answer
28 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
38 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
76 views
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0answers
48 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
votes
2answers
23 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
24 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
38 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
23 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
151 views
0
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1answer
34 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 ...
1
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1answer
12 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
37 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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vote
2answers
36 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. ...
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1answer
16 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, ...
0
votes
1answer
41 views

Birational transformation of Elliptic curves?

Let $F:V\to W$ be a birational transformation of elliptic curves; let $g$ be a generator of $V$. Is necessarily $F(g)$, a generator of $W$?
0
votes
1answer
31 views

Jacobians and ranks of a curve

I would like to know the following: How to find Jacobian and rank of an hyper elliptic curve like $x^5-x= y^2-y$? High regards Rosy
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1answer
33 views

Fourier Series of Eisenstein series [closed]

$$G_{2k}(\tau)= 2\zeta(2k)+2\frac{(2\pi i)^{2k}}{(2k-1)!}\sum_{n\geq 1}\frac{n^{2k-1}q^n}{1-q^n}$$ where $q =e^{2\pi i \tau}$ and $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $L(\tau ...
0
votes
1answer
17 views

Subgroups of points of order 2 in an elliptic curve

Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one ...