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

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Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
4
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1answer
31 views

Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...
4
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1answer
129 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 ...
4
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1answer
70 views

How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
4
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1answer
74 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
4
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1answer
102 views

A question on level structures on elliptic curves

I have a question on $\mathbb{H}/\Gamma(N)$, which parametrizes level $N$ structures on elliptic curves. Let $Y(N)$ be the set of isomorphism classes of such objects, then, according to Fact 2 on page ...
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117 views

Hyperelliptic curve order

How to compute order of a hyperelliptic curve ($y^2=f(x)$, $deg(f)=2 \cdot g+1$, $g=4$), over $F_p$ for small $p$ ($p$ prime)? Are there any efficient algorithms to do so? Is it possible with ...
4
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1answer
180 views

Extend an holomorphic function defined on a torus

Suppose we have an holomorphic function $$ f : \frac{\mathbb{C}}{\Lambda} \mapsto \frac{\mathbb{C}}{\Lambda} $$ where $\Lambda$ is a lattice. Is it always possible to find another function $\psi : ...
4
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1answer
62 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
190 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
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69 views

Other names for $E_{p+1}$ $\pmod{p}$?

If I want to know properties of $E_{p+1}$ modulo $p$, do you know a name for this modular form, so that it is easier to search via the internet? So far, what I know is that $E_{p-1}$ is the Hasse ...
4
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1answer
136 views

Mathematics for Pleasure of a Beginner

I've just read "The Music of the Primes" by Marcus du Sautoy, it is worth a read. I'm not from a maths background, but I'd like to develop a deeper understanding of the concepts. The poetry of math is ...
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1answer
576 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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1answer
112 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 ...
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0answers
121 views

Complex multiplication - Ray class fields

I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin. According to ...
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0answers
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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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2answers
328 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). ...
3
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1answer
757 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 ...
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3answers
180 views

Reference: Elliptic curves as complex tori

I'm looking for books which contain a more or less self-contained description of how elliptic curves over $\mathbb{C}$ - that is, nonsingular plane cubic curves - can be realized as a quotient of the ...
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2answers
382 views

References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
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240 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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3answers
313 views

Rankin-Selberg zeta function

I was reading this paper by de Weger and in conjecture 7 he mentions "the Riemann hypothesis for the Rankin-Selberg zeta function associated to the weight 3/2 modular form associated to E (an elliptic ...
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2answers
299 views

Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime.

Let $Y^2=f(X)$ be an Elliptic curve over a finite field $\mathbb{F}_p$ where $f(X)=X^3+aX+b$ In an undergraduate coursebook on an Applied Algebra course it states that "It is plausible to suggest ...
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683 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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107 views

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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2answers
2k views

How elliptic arc can be represented by cubic Bézier curve?

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
3
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1answer
72 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
3
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1answer
310 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?
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246 views

Isomorphism of Elliptic Curves:

In Stinson's Cryptography Theory and Practice, a theorem is given without proof: Theorem 6.1 Let $E$ be an elliptic curve defined over $Z_p$, where $p$ is prime and $p > 3$. Then there exist ...
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1answer
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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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Genus of Edwards curve

Let us work over a field $\Bbbk$ of characteristic not equal to two. Let $d\in\Bbbk\setminus\{0,1\}$. It is said in the wikipedia article about Edwards curves that the plane quartic defined by the ...
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1answer
118 views

Division polynomials of elliptic curves

This is exercise 3.7 from Silvermans AEC (2nd edition). Let $E$ be a nonsingular elliptic curve over $\mathbb{C}$ given by $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ The $n^{th}$ division polynomls ...
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160 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
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35 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
3
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2answers
62 views

The torsion of an elliptic curve over a finite field

There is a result for $p$ prime, $E$ an elliptic curve over $\mathbb F_p$, then $E(\overline{\mathbb{F}_p})[m]\cong (\mathbb{Z}/m\mathbb{Z})^2$ for $m \nmid p$. The book on cryptography I am using ...
3
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1answer
72 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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4answers
380 views

Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
3
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1answer
137 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
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2answers
61 views

Infinite family of genus one non-elliptic curves over the rationals

How easy is it to write down genus one curves over $\mathbf Q$ without a rational point? Can we write down an infinite family?
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2answers
182 views

Uncertain about Uniformizing Elements of Elliptic Curves.

I am following a subject on Elliptic Curves and have come accross the notion of a uniformizer. Wikipedia tells me that an element is a uniformizer of a Discrete Valuation Ring, if it generates the ...
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2answers
671 views

Point addition on an elliptic curve

I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$. I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive ...
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1answer
93 views

Write an elliptic curve with coefficients only depending on its j-invariant

Let $$E:y^2 = 4x^3-g_2 x - g_3$$ be an elliptic curve and $$j=\frac{g_2^3}{g_2^3-27 g_3^2}$$ denote to its $j$-invariant. I want to transform $E$ to find $f$ and $g$ s.t. ...
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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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the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
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94 views

Ranks of elliptic curves as the base field varies

Suppose that $E/\mathbb{Q}$. Is it the case that for any $r > 0$ there exists a finite field extension $\mathbb{Q} \subset K$ such that the rank of $E/K$ is greater than $r$?
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212 views

Finding a pencil of elliptic curves parametrized by a given modular surface

The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
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1answer
86 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
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Reduction map on torsion of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both ...
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1answer
55 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same ...
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1answer
55 views

Endomorphism Ring of an Elliptic Curve over Finite Field

Let $~E:y^2=x^3+x~$ be an elliptic curve over finite field $\mathbb{F}_{5},$ I compute the trace of Frobenius is $2$($E/\mathbb{F}_{5}$ obvious is ordinary). (By the theory of CM, I know (when $E$ ...