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Questions tagged [elliptic-curves]

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

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Bad reduction at prime numbers for the elliptic curve $y^2+y=x^3-x^2+2x-2$

Consider the elliptic curve $$E:y^2+y=x^3-x^2+2x-2.$$ My goal is to compute the conductor of the elliptic curve, the example is from https://planetmath.org/conductorofanellipticcurve. My problem isn'...
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1answer
13 views

Unramified field extension and elliptic curves

Let $E/K$ be a elliptic curve over a number field $K$ and let $L/K$ be a finite abelian (Galois) extension. Let $v’$ be a (finite) place of $L$ lying over a place $v$ of $K$. Let $I=I_{v’/v}$ be the ...
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1answer
52 views

The 3-torsion Points of an Elliptic Curve over Finite Fields

Let $E/\mathbb{C}$ be a smooth elliptic curve, and let $E[3]$ be the 3-torsion points. We have a non-canonical isomorphism $$ E[3] \cong \mathbb{Z}/3 \times \mathbb{Z}/3$$ and if I'm not mistaken, ...
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1answer
28 views

Torsion points of an elliptic curve (example in Silverman)

Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(\mathbb{Q})_{tors}$ injects into the reduction $\widetilde{E}(\mathbb{...
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66 views

Fermat-Catalan eighth powers

There are two Fermat-Catalan solutions that have as an eighth power in their addend the numbers, $33^8$ and $44^8$. In Darmon and Granville's paper, they show that the generalized Fermat Equation has ...
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1answer
33 views

Is there an elliptic curve mod n with exactly one point?

I have tried many elliptic curves $y^2 = x^3 + ax +b$ with no success. I know that for prime modules there exists a minimum number of points the elliptic curve has to have, and I couldn't satisfy this ...
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1answer
57 views

Does Fermat's Last Theorem imply the modularity theorem?

The Wikipedia article on the proof of Fermat's Last Theorem has this sentence If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either of ...
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26 views

Tangent lines of elliptic curve with a fixed point and Weierstress form

I'm reading Ian Connell's Elliptic Curve Handbook for the details of Nagell's algorithm, which can construct the birational map from an elliptic curve to its Weierstress form. At the bottom of Page ...
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1answer
27 views

What happens to the group structure of an elliptic curve over a field when the discriminant = 0?

Working on a question for a number theory class. So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero? So, basically,...
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1answer
52 views

How can one show that an elliptic curve has a point whose tangent line also meet another point of the curve

I'm coming from a projective setting of a smooth cubic plane curve over a field $K$ and want to show that I can bring it to the Weierstrass long form. The usual method is to start with a point and ...
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Elliptic curve with only one point II

Is there an elliptic curve $E$ over an infinite field $K,\text{char }K\neq0$ such that $E(K)=\{\infty\}$? As noted in the previous question, such a field cannot be algebraically closed, so if there ...
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1answer
49 views

Elliptic curve with only one point

Is there an elliptic curve $E$ over an infinite field $K$ such that $E(K)=\{\infty\}$? My original task was to find an elliptic curve over some field $K$ with only one point, which I did for $K=\...
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46 views

How is the Frobenius morphism of a supersingular curve non-integral if its characteristic equation splits over Z?

Let's take a supersingular elliptic curve $E / \mathbb{F}_q$, where $q = p^2$ for some $p \equiv 1 (\text{mod } 4)$, and $p \geq 5$, let's say. Then $ \# E(\mathbb{F}_q) = (p - 1)^2$, which by Hasse's ...
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1answer
29 views

Non-analytic homomorphism between complex tori

Let $\Lambda_1$ and $\Lambda_2$ be lattices in $\mathbb C$, and let $\phi\colon \mathbb C/\Lambda_1\rightarrow \mathbb C/\Lambda_2$ be an analytic map. Then we know that $\phi$ is a group homomorphism ...
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1answer
47 views

Frobenius map on elliptic curves over a finite field

Let $E: Y^2=X^3+Ax+B$ be an elliptic curve, defined over $\mathbb{F}_p$ where $p$ is a prime. Define: $$\phi: E(\bar{\mathbb{F}}) \rightarrow E(\bar{\mathbb{F}})$$ by $$\phi(P) = \left\{ \begin{array}...
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If the number of points of two elliptic curves are close enough, then they are equal?

$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} $ On page 23 of this paper, one can read: For ...
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20 views

Generalized Elliptic Curves over finite field?

I am interested about the way to classify Generalized Elliptic Curves over the finite field $\mathbb{F}_3$, but i didn't find any references. I wish to understand the techniques used in such ...
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75 views

Find $x$ such that $x^3+ax+b$ is a square mod $p$

Let $p>3$ be a prime, and $a,b \in \Bbb F_p$ such that $4a^3+27b^2 \neq 0$. Can I find some $x \in \Bbb F_p$ such that $x^3+ax+b$ is a square mod $p$ without using Hasse's bound? I know that $\Bbb ...
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Computing Elliptic Curve Isogeny from knowledge of the kernel of the dual Isogeny

Let $E_0(\mathbb{K})$ and $E_1(\mathbb{K})$ be two Elliptic Curves with $char(\mathbb{K})>3$ (e.g. satisfying a Weierstrass-equation). Let $\phi:E_0\to E_1$ be a separable isogeny and $\hat\phi:E_1\...
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72 views

$\alpha$, and the $2L$ conductor in Serre's proof on $a^p+b^p+L^α c^p=0$

In Serre's paper, On the two-dimensional modular representations of $\operatorname{Gal}(\overline{Q}/Q)$, on page 25 Section 4.3, Variants of Fermat's Theorem, Serre gives a proof that $a^p+b^p+L^α c^...
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41 views

Elliptic Curves in Cryptography [closed]

Elliptic curve cryptography is based on finding intersections of lines and elliptic curves: $$y^2 = x^3 + ax + b ~~\text{and}~~ y = ax + b$$ It make sense when you see it on the graph, but the ...
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Elliptic Surface

I can't find a reference for this, so hopefully this isn't too trivial. I know that we have classification of singular fibers by Kodaira but i could't find reference for that. more over I don't know ...
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35 views

Why is it a free module of rank 1?

Let $E$ be an elliptic curve over a field $K$ and $\ell \neq \operatorname{char}(K)$ be a prime, assume that $F := \operatorname{End}_K(E) \otimes_{\Bbb Z} \Bbb Q$ is a quadratic field. In this ...
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1answer
29 views

Endomorphism ring of elliptic curves over $\Bbb Q$ vs over $\Bbb C$

Let $E$ be an elliptic curve over $\Bbb Q$. What is the relation between $End(E)$ and $End(E_{\Bbb C})$ ? We clearly have an inclusion $End(E) \subset End(E_{\Bbb C})$ : given $f :E\to E$, we can ...
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3answers
131 views

Explain these integer solutions

$a+b=u^2, a^2+b^2=v^4$ I have found the solution: $a=4565486027761, \quad b=1061652293520, \quad u=2372159, \quad v=2165017$. But I do not know a more theoretical way to get them.
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31 views

$\Bbb C/(1+ib)\Bbb Z$ is an elliptic curve

I have to prove that $\Bbb C/(1+ib)\Bbb Z$ is an elliptic curve, where $b\in\{1,2,3\}$. The problem is that the teacher was a bit broad giving the definition of elliptic curve. He presented them as ...
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1answer
26 views

If isogenous elliptic curves have equal numbers of points, how can isogenies have non-trivial kernels?

Consider: Silverman, Ex V.5.4: Elliptic curves $E/\mathbb{F}_q$ and $E'/\mathbb{F}_q$ are isogenous if and only if $\#E(\mathbb{F}_q) = \# E'(\mathbb{F}_q)$. Silverman, Proposition 4.12: any finite ...
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1answer
21 views

Determining if an integer is a square in a finite field

Looking at the elliptic curve C: $Y^{2} = X^{3} + X + 2$ in the finite field $\mathbb{F_{13}}$. I first looked at the line $X = 2$, which yields $Y^{2}= 12$, and was "lucky" and saw that $12+13 = 25 =...
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1answer
57 views

Elliptic curve with only torsion points

Is it possible to have an elliptic curve $E$ over $\Bbb Q$ such that $E( \overline{\Bbb Q})$ is a torsion abelian group? I know that $E(\Bbb Q)$ can be a finite group. I know that $E(\Bbb C)$ is a ...
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1answer
72 views

Division polynomial of a super-singular vs ordinary Elliptic Curve

For an Elliptic Curve in Finite Field of characteristic $p$, I'm trying to understand how the division polynomial for multiplication by field-characteristic differs between an ordinary curve and a ...
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Modularity and the most general $(p, q, r)$ case in the BeChDaYa paper.

In the Bennet, Chen, Dahmen, Yazdani paper, Generalized Fermat equations: A miscellany, on page 24 in section 6 entitled "Future Work", they say: "A limitation of the modular method at present is ...
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1answer
23 views

Rational points on elliptic curve over a local field correspond to integral points on minimal Weierstrass model

Let $R$ be a complete discrete valuation ring, with fraction field $L$. Let $E$ be an elliptic curve over $L$, and $W$ a minimal Weierstrass model over $R$. Why is $W(R) \simeq E(L)$? We have a map $...
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Showing two tori are non-isomorphic as complex manifolds

I want to show that there exist two elliptic curves over $\Bbb C$ that are non-isomorphic. Let us write $\Gamma_1=\Bbb{Z}\oplus \Bbb{Z}\tau_1$ and $\Gamma_2=\Bbb{Z}\oplus\Bbb{Z}\tau_2$ where $\tau_1,\...
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Is there a “Weierstrass form” for isogenies?

Let $W_1$ and $W_2$ denote smooth subsets of $\mathbb{P}^2$ given by Weierstrass equations and suppose $\varphi : W_1 \rightarrow W_2$ is an isogeny. Is there a specific form that $\varphi$ can be ...
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1answer
67 views

Deriving Elliptic Curve Addition Formula

I just have a very basic question. For two distinct points $P_1, P_2$ with non-zero components on an elliptic curve $C$ given by $y^2 = x^3 + Ax + B$, I'm trying to derive the addition formula using ...
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68 views

Elliptic curve with same number of point over two different fields

Following a discussion with a number theory professor, we arrived at the following question : Can we find an elliptic curve (short form : $y^2=x^3+ax+b$) with an identical number of points on two ...
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How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?

It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote. It would be cool because we could exploit $\sum_{n=1}^N\...
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When do compositions of isogenies of elliptic curves respect isomorphisms?

Let $E/ F_p$ be an elliptic curve. Let $G_A$, $G_{A'}$, $G_B$ and $G_{B'}$ be subgroups of $E(\overline{F_p})$. Suppose $E/G_A \simeq E/G_{A'}$ and $E/G_B \simeq E/G_{B'}$. When is $E/<G_A,G_B> \...
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33 views

General formula for the probability to find a factor in the elliptic curve method?

I am aware of tables showing the optimal $B1$ and the probabilities for some values. But I would like to know the general approach to get the probabilities. Suppose, a given factor with $n$ digits ...
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Formal completion of modular curves

Let $N\geq 4$ be an integer coprime with $p$, where $p$ is a fixed prime number. Then we know that there exists a scheme $Y_N$ over $\text{Spec}(\mathbb{Z}_p)$ whose $\text{Spec}(R)$-points are ...
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Counting Homogeneous Forms on Elliptic Curves

I'm trying to understand some argument that is being made regarding homogeneous forms on an elliptic curve. In order to do this, I need to understand how we determine the dimension of the space of ...
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3answers
52 views

The parametrisation (cos(t),sin(t+a)) almost always seems to describe a rotated ellipse. How can I prove it?

I was wondering how a curve would look like if you have two hands of a clock moving at the same speed from different positions while you track the x of one hand and the y of the other hand over time. ...
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Is there an upper bound to the number of weight $2$ newforms for any level $N$?

There are no newforms of weight $2$ for levels $1$ thru $10,12,13,16,18,22,25,28,60$. Looking at the list given in Sloane's integer sequence A127788 it appears that the number of newforms increases ...
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Does the elliptic curve $y^2 = 4 x^3 -6075$ have any integer points?

Let $E$ be the elliptic curve $y^2 = 4 x^3 -6075$. I ran the following Mathematica code, which searches naively for integer solutions to $E$ but it did not find ...
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2answers
47 views

Polynomial curves $F(x,y)$ giving rise to loops

I know that the cubic curve defined by $F(x,y) = y^2 - x^3 - x^2$ gives rise to a loop: Is there a way to check if a cubic curve defined by $F(x,y) = y^2 + ay + bx^3 + cx^2 + dx + e$ exhibits a loop? ...
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54 views

Example of addition of Twisted Edwards Curve on Sage, Python,..

Can somebody tell me how can I make a example of addition in Twisted Edwards Curve on sage? For example: $ax^2 + y^2 = 1 + dx^2y^2$ Given the following twisted Edwards curve with $a=3$ and $d=2$: ...
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2answers
84 views

How do you construct the Frey curve for (2,3,p)?

In Darmon's paper on p.14 he lists a table of signatures $(p,q,r)$ and constructed Frey curves. How do you construct the Frey curve he gives for $(2,3,p)$? The curve he gives for this signature is: ...
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81 views

Cremona 2.14.1 Why is $c_4$ and $c_6$ complex when they should be rational?

In Cremona's online book Chapter 2, in order to calculate the lattice invariant we have: $\tau=\omega_1/\omega_2$ Set $q=e^{2\pi i\tau}$ (2.14.1) $c_4 =(2\pi/\omega_2)^4(1+240 \sum_{n=1}^{\infty} n^...
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1answer
26 views

Stage 1 of Elliptic Curve Method (ECM)

Reading several texts of ECM (e.g. 20 years of ECM) the Stage 1 is described as: $Q \leftarrow P_0$ for each prime $\pi <= B_1$ $\quad$compute $k$ such that $\pi^k <= B_1 < \pi^{k+1}$ $\quad$...
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68 views

Can the coefficients of an elliptic curve be recovered from the curve itself?

Let $K$ denote an algebraically closed field. Suppose $E$ and $E'$ are elliptic curves given by Weierstrass equations with parameters $(a_1,a_3,a_2,a_4,a_6)$ and $(a_1',a_3',a_2',a_4',a_6').$ From $E =...