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

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Pictures of curves over finite fields with many points

At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$: $(x + y +...
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Weiertrass equation of an elliptic curve.

We know that every elliptic curve is a non-singular $\textbf{cubic}$ projective curve (curve of genus 1), but we can transform this in the Weiertrass form $$y^2 + a_1xy + a_3y = x^3 + a_2x^2 +a_4x +...
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Special Euler bricks and $x^2(y^2-1)^2+y^2(x^2-1)^2=z^2$

Define, $$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$ Let, $$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$ and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then, $$P_3:=x^2(...
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Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
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What happens in elliptic curve primality testing if you cannot find a suitable discriminant?

I'm trying to understand the computational aspect of elliptic curve primality testing (specifically the Atkin-Morain test), and in general, I understand why it works for a prime number. However, when ...
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Discriminant of an elliptic curve

I have found different discriminants for general Weierstrass elliptic curves: $y^2 = x^3 + ax + b$ For example, WolframAlpha state it is $-16(4a^3 + 27b^2)$ on their site http://mathworld.wolfram.com/...
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How to find the equation of the curve defining the intersection of two quadrics.

Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that $\lambda\...
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How to find points at infinity in projective coordinates for the $x^2 -3xy-2y^2-x+1=0$

How to find points at infinity in projective coordinates for this algebraic curve $x^2 -3xy-2y^2-x+1=0$ I don't know how to start! Any help will be appreciated.
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How to prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$

Let $f(x) = x^3 +ax+b$ contained in $Q[x]$ prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$ This is what I've so far since $f(x) = (x-A_1)(x-A_2)(x-A_3), A_1,A_2, A_3$ are ...
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What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
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Discriminant of Elliptic Curves

In the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = -16(...
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Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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Order of a point on elliptic curve

I am trying to prove that the following elliptic curve has rank=1: $$y^2=x^3+x^2+x+1$$ From the map $$\delta: E(Q)\rightarrow Q(i)^*/(Q(i)^*)^2$$ $$(x,y)\mapsto (x-i) $$ for $(x,y)\neq O$ and $O\...
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Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of $\mathbb{P}^2(\...
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Divisor on curve of genus 2

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points $...
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Computing the dual of frobenius endomorphism

There is an exercise in my course Elliptic Curves and I am not sure if I am doing it right. The question is as follows: Let $E$ be the elliptic curve over $\mathbb{F}_2$ given by $Y^2+Y=X^3$. (a) ...
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What is the right dual isogeny?

I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of unity....
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Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and "...
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Elliptic Curves over Finite Fields as Two Cyclic Groups

Let $E$ be an elliptic curve over $\mathbb{F}_q$. I want to show $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ where $m_1,m_2 \in \mathbb{Z}$ are such that $...
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Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
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How to convert $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$

I'm to figure out how to convert algebraic curve such as $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$. Any help will be appreciated!
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Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
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Group law on elliptic curves.

Let $k$ perfect field. If we have a cubic non-singular projective curve $C(k)$ (over a field $k$), take two diferent points $P_1,P_2 \in C(k)$ and consider the line through the points, by Bezout ...
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Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I've been attempting to prove some comments I've read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne's ...
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Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take $\mathbb{P}^{1}_{\mathbb{...
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Property of Weierstrass sigma function

In theorem 1.2.3 of Schertz' Complex Multiplication says that For any $\omega \in \mathcal{L}$, a fixed lattice, we have the property: $$ \sigma(z + \omega) = \psi(\omega)e^{\eta(\omega)(z + \omega/...
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General theory of discriminants

I read it in the book Rational Points on Elliptic Curves by Silverman and Tate: If $f(x)$ is a polynomial with leading coefficient 1 in $Z[x]$, then the discriminant of $f(x)$ will be in the ideal ...
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Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
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Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free $A$-...
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Lifting a real quadratic twist of an Elliptic Curve to the modular surface

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ ...
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Isogeny of elliptic curves over $p$-adic field

If $K$ is a $p$-adic field, and $E_q$ and $E_{q'}$ are the corresponding Tate curves for $|q|,|q'|<1$, why does $E_q$ and $E_{q'}$ being isogenous imply that there are integers $A$ and $B$ such ...
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Help with getting the formula for rational point $(x,y)$ on the $y^2 = x^3$

How to find formula for rational point $(x,y)$ on the $y^2 = x^3$ in term of rational parameter $t$. And also how would I write in form of $(x,y) =(f(t),g(t))$
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Finding formula for rational point ($x$,$y$) on the y = $x^2$

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))
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example of an elliptic curve without complex multiplication

What is an example of an elliptic curve $E$ without complex multiplication? This means $End(E)=\mathbb{Z}$. I know that complex elliptic curves are given by $\mathbb{C}^2/\Lambda$ for a lattice $\...
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Elliptic curves over $\mathbb{Q}$, singularity points

Why is it so that $Y^2Z = X^3 + AXZ^2 + BZ^3$ is a non-singular elliptic curve if $4A^3 - 27B^2 \neq 0$? If we check the partial derivatives we get that $\frac{\partial F}{\partial Z} = Y^2 - 2AXZ - ...
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Sum of two cubes transformed to elliptic curve

Given $x^3+y^3=N$, we can perform some substitutions to obtain an elliptic curve $u^3-432N^2=v^2$, as given here, which are $x=\frac{36N+v}{6u}$, $y=\frac{36N-v}{6u}$. Here's the details: \begin{align*...
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Change of coordinate in Weistrass Normal form

From Rational Points on Elliptic Curves by Silverman and Tate To obtain Weistrass Normal form, it uses some technique of changing coordinates, but I can't understand its description, I underlined ...
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How to prove that $\frac{1}{n}L/L\simeq (\mathbb{Z}/n\mathbb{Z})^2$?

Let $L$ be any lattice in $\mathbb{C},$ and $L'$ a lattice containing $L$ with index $n$ (i.e $n=\sharp L'/L$) I found this statement "The lattice $L'$ must be contained in $\frac{1}{n}L = \{\...
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Surjectivity of morphisms of smooth projective varieties

I have a question regarding a proof of the "surjectivity of morphisms of projective varieties" (a whole mouthfull). Though there are proofs using completeness of varieties, I am interested in an ...
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Nagell-Lutz theorem

I'm a question about Nagell-Nutz theorem. For example, the point $P'=( \frac{31073}{2704},-\frac{5491823}{140608})$ on the curve $$y^2=x^3+8$$ does not meet the theorem Nagell-Nutz. So I can say what ...
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J. Silverman exercise 3.12 “The arithmetic of Elliptic curves”

I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following: Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the ...
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Weierstrass form of elliptic curve with point with order larger than 3

L.S., Studying for my exam on elliptic curves, I tried to make exercise 8.13(a) of Silvermans "The Arithmatic of Elliptic Curves", which reads: Let $E$ be an elliptic curve defined over a field $k$, ...
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Maps between Elliptic Curves and Points at Infinity

I was trying some exercises from Silverman's book Rational Points on Elliptic Curves 2nd ed. (2015), and got stuck at this problem. 1.22 Let $C$ and $W$ be the projective curves ($b,e \ne 0$) $$ C: ...
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When is sum of squares a perfect square? [duplicate]

Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding ...
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sheaf of relative differentials on an elliptic curve

Let $f : E\rightarrow S$ be an elliptic curve over a scheme $S$ with identity section $e : S\rightarrow E$. Why is it true that $e^*\Omega_{E/S}\cong f_*\Omega_{E/S}$? (I believe these should be ...
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Properties of the elliptic curve $y^2 \equiv x^3 – 2 \pmod 7$

Can someone help me: 1) to list the points on the elliptic curve $E: y^2\equiv x^3 – 2\pmod 7$. 2) to find the sum $(3, 2) + (5, 5) $ on $E$.
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Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
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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 ...