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

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3
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38 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
0
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1answer
21 views

Order's size (in bits) of an elliptic curve

I am trying to prove that, given an Elliptic Curve defined on $\mathbb{F}_p$ with $p$ a prime number, the order $q$ verifies: $|p| \le |q| \leq |p|+1$ where $|x|$ denotes the length in bits of $x$. ...
4
votes
2answers
79 views

Rational solutions to $x^4+y^4=cz^2$

Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve? What I really want to know is how to transform this into an elliptic ...
2
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0answers
91 views

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
0
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1answer
37 views

The sum of an isogeny and its dual for the Frobenius homeomorphism

This is from page 150 of Silverman's "The Arithmetic of Elliptic Curves". Any my only questions is: How you can conclude that $[a]=\phi+\hat{\phi}$? I tried to use the formula on page 85 which ...
0
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0answers
47 views

Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
4
votes
1answer
100 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
2
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0answers
43 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
2
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0answers
65 views

Isogenies between elliptic curves with specified torsion groups

For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert ...
2
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1answer
25 views

Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
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2answers
86 views

Arithmetic on modular curves

I had tried to read the first few pages of Glenn Stevens' Arithmetic on Modular Curves, but it is somehow extremely unreadable to me, the text format is odd and stating too much facts without ...
2
votes
1answer
80 views

Solve for $x$ in elliptic curve $y^2 = x^3 + ax + b$

Given $y$, is it possible to solve for $x$ in the elliptic curve equation $y^2 = x^3 + ax + b$ over a finite field? Or is it known to be as difficult as say, something like the discrete logarithm ...
1
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1answer
40 views

Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
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0answers
28 views

How to find the affine coordinate algebra of $n$-torsion points of elliptic curve?

In particular, I'm wondering about the affine coordinate algebra of $E[3]$ where $E$ is the elliptic curve $y^{2}=x^{3}-D$ over $\mathbb{Q}$ with $D=2^{8}3^{5}5^{2}$. I think we can view $E[3]$ as ...
3
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0answers
52 views

The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
0
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1answer
16 views

Addition of x-coordinate on elliptic curve given by Möbius Transformation

Consider the elliptic curve $y^2=(x-\alpha)(x^2+ax+b)=x^3+(a-\alpha)x^2+(b-a\alpha)x-\alpha b$ over the field $K$ with $\text{char}\ K\not= 2$. The questions I am doing asks for a formula for the ...
0
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2answers
72 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
3
votes
1answer
32 views

Galois Representation with $D_{10}$ image

I want to construct an explicit elliptic curve $E$ over a number field $K$ such that $Gal(K(E[l])/K) \cong D_{10}$ where $D_{10}$ is the dihedral group of order 10 and $l$ is a prime number. ...
4
votes
0answers
93 views

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is. One of the ...
0
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1answer
28 views

can someone explain Nagell-Lutz theorem

(elliptic curve $y^2 = x^3 + ax^2 + bx + c$) Nagell-Lutz theorem: If $p(x, y)$ is finite order on a given integer coefficient elliptic curve satisfy: (1) x and y are integer (2) y = 0 or y | D (D ...
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5answers
357 views

upper bound on rank of elliptic curve $y^{2} =x^{3} + Ax^{2} +Bx$

I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies $r \leq \nu (A^{2} -4B) +\nu(B) ...
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0answers
28 views

Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
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vote
2answers
26 views

Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or ...
2
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1answer
33 views

height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
4
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0answers
89 views

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

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
6
votes
2answers
86 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 ...
4
votes
1answer
88 views

Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
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3answers
5k views

How could I calculate the rank of the elliptic curve $y^2 = x^3 - 432$?

The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$. Using pari/gp we can compute the ...
5
votes
0answers
109 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
3
votes
1answer
641 views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
3
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1answer
56 views

Calculating Elliptic Curve cofactor h

An Elliptic Curve in short Weierstrass form over a finite field $F_p$ is given by the equation: $$y^2 = x^3 + ax + b \mod p$$ To use this curve for cryptographic purposes, in the domain parameters ...
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vote
1answer
44 views

Show that Weierstrass function is elliptic function.

Prove that Weierstrass function is periodic with respect to lattice $L (L\subset \mathbb{C})$ .i-e $f(z+w,L)=f(z,L)$ ($w\in L$). $f(z,L)=\frac{1}{z^2}+\sum_{0\ne w\in ...
2
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1answer
26 views

Point conversion between Twisted Edwards and Montgomery curves

With the great help of Birational Equvalence of Twisted Edwards and Montgomery curves I know how to convert twisted Edwards curves into their birationally equivalent Montgomery counterparts where I ...
4
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2answers
54 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
9
votes
1answer
132 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
2
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1answer
29 views

Birational Equvalence of Twisted Edwards and Montgomery curves

I'm trying to understand the birational equivalence between Twisted Edwards and Montgomery curves and try to calculate some examples. In particular, as an example, I'm looking at the Ed25519 Twisted ...
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59 views

Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
3
votes
1answer
37 views

Group operations on Montgomery Curves in affine representation

I'm trying to understand group operations on elliptic Montgomery curves in affine representation. Let's say the curve I use is Curve25519, i.e.: $$y^2 = x^3 + A\,x^2 + x\quad\text{where}\quad ...
3
votes
1answer
40 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} = ...
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votes
2answers
126 views

Homogeneous diophantine equation $x^3+2y^3+6xyz=3z^3$

Is it known if there are infinitely (non-proportional) many integer solutions to $x^3+2y^3+6xyz=3z^3$ ? Motivation : if true, this would provide an alternative solution to that recent MSE question, ...
3
votes
1answer
79 views

Why do the Diophantine equation $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=n$ gives an elliptic curve?

In a book "Which way did the bicycle go" was tought a problem of integer solutions of certain Diophantine equation. This is the idea, not an exact quotation: For which integers $n$ are there integers ...
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0answers
109 views

What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
3
votes
1answer
71 views

Definition of Selmer-Group for Elliptic Curves

Im facing a problem in Silvermans Book "Arithmetic of elliptic Curves" at the beginning of chapter X.4 concerning the exact sequences. Let $K$ be a number field with a valutaion $v$. I'm ...
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0answers
26 views

Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil $L$ function. Is the rank of the abelian group $E(K)$ of points of $E$ the ...
2
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1answer
166 views

Modularity theorem and some results

Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem http://en.wikipedia.org/wiki/Modularity_theorem In the section ...
7
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1answer
204 views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic ...
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0answers
32 views

Weierstrass-$\wp$ Function Asymptotics

Given the Weierstrass-$\wp$ function, $$\wp(2x+1+\tau \mid 1, \tau),$$ with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...
5
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0answers
231 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
3
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0answers
47 views

Are there any other integer points on the elliptic curve $Y^2 = X^3 + 1$ beyond $(-1, 0), (0, \pm 1), (2, \pm 3)$?

The charm of elliptic curves is that given one or two integer points, one can find others by the group law. However the easy to guess points from the title just pump me around trough a cyclic group of ...
4
votes
2answers
50 views

Third point of intersection is also a point of inflection?

Let $C \subset \mathbb{P}_2$ be a nonsingular cubic. If $L$ is a line through two distinct points of inflection on $C$, how do I show that the third point of intersection is also a point of ...