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

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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 ...
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1answer
255 views

seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\Spec}{\text{Spec }}$ It's a well known fact that every elliptic curve (say, over a field $k$) has a global holomorphic nowhere vanishing differential. If ...
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1answer
250 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
762 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 ...
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1answer
214 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. ...
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1answer
148 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 ...
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1answer
140 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 ...
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72 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 ...
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1answer
126 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 ...
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Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
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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 $E$, ...
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1answer
134 views

Automorphisms of elliptic curve

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?
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1answer
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How to construct isogenies between elliptic curves over finite fields for some simple cases?

From the theorem of Tate, it is known that the two elliptic curves over the same field $\mathbb{F}_p$ are isogenous iff they have the same number of points. For $p\equiv 3\mod 4$, the curve ...
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1answer
104 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 ...
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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 ...
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1answer
503 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 ...
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1answer
128 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 ...
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175 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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1answer
935 views

Intersection of a line with an Elliptic Curve

I am trying to show that if a line given by $y = mx + b$ intersects an Elliptic Curve given by $E(\mathbb{K}): y^2 = x^3 + Ax + B$ in three points then the line is not tangent to the curve. Given ...
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Canonical sheaf of bielliptic surface

This is an example of bielliptic surface on page 84 from Beauville's book "Complex algebraic surfaces". Let $\rho^3=1$, $\rho\neq1$ and $F_\rho=\mathbb{C}/(\mathbb{Z}+\rho\mathbb{Z})$, ...
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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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Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
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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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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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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 ...
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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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143 views

Complex Multiplication/ Elliptic Curves Question

I'm working on the following problem from Silverman's advanced topic in the arithmetic of elliptic curves: Let $E$ be an elliptic curve defined over a number field $L$ with complex multiplication by ...
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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 ...
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446 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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157 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 $C\hookrightarrow E\times E$ ...
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1answer
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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 ...
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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 $p$, ...
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507 views

Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points

Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve. By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$. By implicit ...
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103 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 $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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600 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 ...
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2answers
355 views

Rational points on an elliptic curve

Consider the following elliptic curve $y^2=(x+1540)(x-508)(x-65024)$. It is trivial that the points $P_1(-1540,0)$, $P_2(508,0)$ and $P_3(65024,0)$ lie on this curve. It is also quite easy to find ...
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996 views

Finding zeros and poles of rational functions over elliptic curve

The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point $(0, 1, 0)$) still seems unclear. For example, Silverman ...
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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 ...
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1answer
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Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
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1answer
435 views

Square root in Characteristic 2 Field

Let $K$ be a field of characteristic 2. For each $a\in K$, can we always find some $x$ such that $x^2=a$? I came upon this question while reading "Arithmetic of Elliptic Curves". The original ...
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480 views

Question about quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $d$ be a squarefree integer. If $E'$ and $E$ are isomorphic over $\mathbb{Q}(\sqrt{d})$, must $E'$ be a quadratic twist of $E$?
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Example of an elliptic curve with trivial torsion subgroup and rank 0

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?
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199 views

Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb ...
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364 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
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1answer
65 views

Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
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101 views

Galois representations and isogenies of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a ...
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Concrete and elementary applications of modular forms to elliptic curves

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery? It's often been asked what elementary ...
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1answer
160 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
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Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...