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

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Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
6
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0answers
140 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
5
votes
3answers
298 views

The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

The elliptic curve, $$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$ has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets ...
5
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1answer
400 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
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4answers
344 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) ...
5
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2answers
512 views

Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?

In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at ...
5
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1answer
234 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
5
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1answer
563 views

Number of 3-torsion points on an elliptic curve

If we take our elliptic curve over $K$ to be the zero set of $$ F(X_1, X_2, X_3) = X_2^2 X_3 - (X_1^3 + AX_1X_3^2 + BX_3^2), $$ which is in projective form with $X = X_1, Y = X_2, Z=X_3$, then I ...
5
votes
2answers
1k views

What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition. For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...
5
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2answers
335 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
5
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1answer
470 views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
5
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1answer
72 views

Exact rank of Elkies curve

A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously). ...
5
votes
2answers
67 views

Property of the $p^n$-Selmer group

Consider the $p^n$-Selmer group of an elliptic curve $E/\mathbb{Q}$, $\operatorname{Sel}_{p^n}(E)$. Must we always have $\operatorname{Sel}_{p^n}(E) \cong (\mathbb{Z}/p^n\mathbb{Z})^{s}$ for some $s$? ...
5
votes
2answers
560 views

Prove that the equation $y^2=x^3-73$ has no integer solutions

Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
5
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2answers
137 views

Computing rank using $3$-Descent

For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that $$ 0 \rightarrow ...
5
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1answer
90 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
5
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1answer
53 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
5
votes
2answers
97 views

How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
5
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1answer
148 views

Discriminants and Weierstrass form of elliptic curves

I'm confused by what appears to be contradictory information. In this post, the claim is made that "Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^{2}= x^{3}+ax+b$ where ...
5
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1answer
158 views

Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
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2answers
330 views

Relation involving the conductor of an elliptic curve

Consider an elliptic curve $E: y^{2} = x^{3} + ax + b$. Then the quadratic twist by a squarefree $d$ is given by $E^{d} : dy^{2} = x^{3} + ax + b$. What is the relationship between the conductor of ...
5
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1answer
176 views

Group structure of an elliptic curve

Let $E$ be an elliptic curve over field $\mathbb{Z}/p\mathbb{Z}$. The curve group $E(\mathbb{Z}/p\mathbb{Z})$ is always a) cyclic or b) direct product of two cyclic groups. First question: How do I ...
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1answer
684 views

How do I show that this curve has a nonsingular model of genus 1?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...
5
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1answer
180 views

Family of elliptic curves with trivial torsion

I'm wondering, if it is true that the torsion subgroup of $y^2=x^3+p$ (for $p$ some prime, greater than 2), is always trivial?. I was trying to prove this using Lutz-Nagell, but I can't quite get it. ...
5
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1answer
133 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
5
votes
1answer
126 views

Complex elliptic curve for the “conjugate” lattice

Let $\Lambda$ be a lattice in $\mathbb{C}$, and $E=\mathbb{C}/\Lambda$ the corresponding complex elliptic curve. Let $\bar{\Lambda}$ be the "conjugate" lattice, i.e. the one obtained by conjugating ...
5
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1answer
70 views

Minimal degree of a field extension to obtain an elliptic curve

Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$ of genus $1$. There exists a number field $L/K$ such that $X$ has a $L$-rational point. Let $L$ be ...
5
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1answer
121 views

Cokernel of morphism of Tate module of elliptic curves

Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell ...
5
votes
2answers
75 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 ...
5
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1answer
166 views

Lenstra's Elliptic Curve Algorithm

I am currently trying to understand Lenstra's Elliptic Curve Algorithm for factoring integers. As a source I use "Rational Points on Elliptic Curves" by Joseph H. Silverman and John Tate. They ...
5
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1answer
103 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
5
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1answer
88 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 ...
5
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1answer
54 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 ...
5
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1answer
382 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 ...
5
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1answer
122 views

Show that an ideal is unramified

See Advanced Topics in elliptic curves for the full question(see also errata: http://www.math.brown.edu/~jhs/ATAEC/ATAECErrata.pdf): 2.30 (pg 184) Given $E/L$ an elliptic curve with complex ...
5
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1answer
148 views

Abelian Elliptic Surfaces

By abelian surface we mean a 2-dimensional algebraic complex torus. Thus $$ S=\Bbb{C}^2/\Gamma$$ where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
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60 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} = ...
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0answers
49 views

Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
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0answers
53 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
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0answers
135 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 $k$, ...
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0answers
145 views

Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a ...
5
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0answers
126 views

Why is this a characterization of isogenies of elliptic curves? (From Silverman)

In the proof of Theorem III.6.2 (c) in Silverman's The Arithmetic Of Elliptic Curves it says: Let $x_1, y_1 \in K(E_1)$ and $x_2, y_2 \in K(E_2)$ be Weierstrass coordinates. We start by looking at ...
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2answers
375 views

On the relationship between Fermats Last Theorem and Elliptic Curves

I have to give a presentation on elliptic curves in general. It does not have to be very in depth. I have a very basic understanding of elliptic curves (The most I understand is the concept of ranks). ...
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3answers
2k views

Integral points on an elliptic curve

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
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2answers
148 views

Subvariety of Product of Elliptic Curves

This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ ...
4
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1answer
1k views

What is the Birch and Swinnerton-Dyer Conjecture?

This is probably a really silly question, but I was wondering if someone could explain the Birch and Swinnerton-Dyer conjecture to me in a simple way. I've read a lot about it, but cannot understand ...
4
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2answers
112 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 $p$, ...
4
votes
1answer
605 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
4
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2answers
91 views

Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
4
votes
4answers
510 views

Elliptic curves with finitely many rational points

A conjecture by Goldfeld says that half of all elliptic curves have rank zero (i.e. their Mordell-Weil group has finite order.) Are there any known infinite families of elliptic curves (over ...