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

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3
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1answer
72 views

Unramified at certain places and Selmer groups

Consider the $p$-Selmer group of an elliptic curve $E/\mathbb{Q}$ denoted by $\operatorname{Sel}_{p}(E/\mathbb{Q})$. Why does showing that $E(\mathbb{Q}_{\ell})/pE(\mathbb{Q}_{\ell}) = 0$ (for $\ell ...
1
vote
1answer
470 views

Elliptic curves mod p

I am currently revising for an exam and need some help with a question. Below is an example from my notes which I am trying to understand. I can fill out the table just fine, but I can't figure out ...
1
vote
2answers
71 views

What does the $[(0 : 3 : 1)]$ means in Sage.

I tried to solve the integer points of $y(y+2)=x^3+(x+3)(x+5)$ by using Sage's command E.integral_points(). Its output was $[(0 : 3 : 1)]$. I tried that ...
1
vote
0answers
68 views

Elliptic curve terminology confusion

I've been reading a paper that says "Let $E(K)$ be an elliptic curve..." where $E(K)$ means the $K$-rational points of $E$ (where $K$ is a number field). I've seen phrases like "Let $E/K$ be an ...
4
votes
1answer
133 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 ...
1
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0answers
158 views

Computing the degree of an isogeny

Let $E$ be an elliptic curve with a $p$-torsion point. Denote this point by $P$. Why does is isogeny $\phi: E \rightarrow E/\langle P \rangle$ of degree $p$? I do know that if $\phi$ is separable, ...
3
votes
2answers
161 views

Property of elliptic curves with a torsion point

Let $E/\mathbb{Q}$ be an elliptic curve with a $p$-torsion point. Does this imply that $E/pE$ is isomorphic to $E$? If not, are there any conditions I can assume such that this is true?
0
votes
0answers
76 views

Moduli space of elliptic curves with $C_n$ action

I would like to construct moduli space of elliptic curves with cyclic group $C_n$-action. In other words, I want to classify a pair $(C,\phi)$, where $C$ is an elliptic curve and $\phi:C_n\rightarrow ...
2
votes
1answer
207 views

Help in finding curve equation.

What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the ...
4
votes
0answers
136 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 ...
2
votes
0answers
51 views

Different elliptic curves over given $\mathbb{F}_q$ can have different orders?

As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders? I assume yes. Let $q:=5$. Wikipedia gives $9$ points for ...
2
votes
1answer
149 views

Nef divisors on the compactified modular curve level $N$

Consider the compactified modular curve with full level structure $X=\overline{\Gamma(N)\setminus \mathcal{H}}$. We know the Hodge bundle (the extension of the hodge bundle to the compactification) ...
5
votes
2answers
570 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.
3
votes
0answers
131 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
-1
votes
1answer
53 views

Does 0 lie on elliptic curve?

Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?
3
votes
1answer
50 views

Conditions such that $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})\not\equiv 0 \bmod{p}$

Let $E/\mathbb{Q}$ be a semistable elliptic curve. Let $\ell$ be a prime of multiplicative reduction and consider $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})$. Given a prime $p \neq \ell$, are there any ...
5
votes
0answers
138 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 ...
1
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0answers
81 views

Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
0
votes
1answer
141 views

Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.

Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points. Using Fermat's ...
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0answers
82 views

Elliptic curves with form$ y^2$=$x^3$+$p^2$$x$

We Know that from a conjecture by Goldfeld says that half of all elliptic curves have rank zero. Are there any known infinite families of elliptic curves in form $y^2=x^3+p^2x$ where p is prime with ...
2
votes
0answers
120 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
2
votes
1answer
136 views

Prove that there are no natural numbers, $i, j$ such that $ 3i^2+3i+7=j^3$

I'm not sure if this is true but, I've tried with many different values of $i, j$ and didn't get any contradictions. The question again, here Prove that there are no natural numbers, $i, j$ such ...
2
votes
1answer
149 views

A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module. Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in ...
5
votes
2answers
343 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 ...
-1
votes
1answer
221 views

Understanding whether a ramified covering ramifies over infinity

Let $C$ be the (smooth) curve in $\mathbb{C}^2$ defined by $y^2 = x^4 - 1$, and let $\pi : C \to \mathbb{C}$, $\pi(x,y) = x$. $\pi$ is a ramified cover, ramified over $\pm 1, \pm i$. $C$ is a ...
3
votes
1answer
236 views

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 $E_B$ ...
0
votes
1answer
106 views

Solutions of $a^2 = b^d -3^c$

The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied ...
2
votes
1answer
318 views

writing down the minimal discriminant of an elliptic curve

Let $j$ be an integer. Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way? For example, can ...
0
votes
0answers
123 views

Pell type equation cum elliptic curve equation

I have seen this equation $y^3 - 3x^2 = p^m$ to determine the solutions. I know this is elliptic curve. I had some knowledge of elliptic curve. But, I was totally upset to determine the solutions of ...
4
votes
0answers
118 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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vote
0answers
69 views

Is there any Elliptic Curve algorithm equivalent to RSA's asymmetric encryption?

I've been searching but I cant find anything about this... only EC Diffie-Hellman with symmetric cryptography, which is exactly what I do not want :( Imagine this: generate a random private key, ...
3
votes
2answers
103 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 $r$?
26
votes
1answer
2k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
8
votes
0answers
117 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
2
votes
1answer
171 views

Complex tori as elliptic curves

I have a question about the proof of the following theorem: A complex torus is conformally equivalent (so isomorphic as Riemann surface) to a complex elliptic curve I used the book "N.Koblitz, ...
2
votes
2answers
126 views

Elliptic curve condition on coefficients

I am working something where a picture like this one appeared : Say the curve is written in the form $$ y^2 = x^3 + ax^2 + bx + c $$ (if this is the wrong form of coefficients, feel free to correct ...
3
votes
0answers
135 views

when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?

If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
7
votes
1answer
302 views

Is the pushforward of the sheaf of differentials on an elliptic curve over a scheme necessarily trivial?

If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section ...
2
votes
0answers
77 views

Galois action on CM elliptic curves

Let $E$ be an elliptic curve $E$ defined over a number field $K$ such that $E$ has complex multiplication by the maximal order in the ring of integers of an imaginary quadratic field $F$. Let ...
6
votes
2answers
539 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 ...
2
votes
1answer
80 views

Tate models and subgroups of type$(m,m)$ - re Silverman, Hindry 88

I need help in understanding a passage in a paper by Hindry and Silverman, "The Canonical Height and Integral Points on Elliptic Curves". (re. page 439) Let $ E(K) $ be an elliptic curve with ...
8
votes
3answers
798 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 ...
9
votes
2answers
399 views

Concrete Example of the Birch and Swinnerton-Dyer Conjecture

The Setup Consider an elliptic curve $E$ in Weierstrass form $y^2=x^3+ax+b$ with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set $N_p := $ ...
4
votes
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 ...
5
votes
1answer
616 views

Proving Fermat's Last Theorem (easily) using “assumed” conjectures

It can easily be proven assuming Szpiro's conjecture that Fermat's Last Theorem is true for sufficiently large $n$. The proof consists of extremely straightforward computations. My question is, is ...
1
vote
1answer
193 views

Drawing elliptic curve

Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional ...
4
votes
2answers
380 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). ...
4
votes
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
votes
3answers
322 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 ...
3
votes
3answers
112 views

Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$.

Let $p(x) = ax^3 + bx^2 + cx + d$ where $a,b,c,d \in\mathbb{R}$. Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$. ...