Questions tagged [elliptic-curves]
For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
1
vote
0answers
29 views
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'...
1
vote
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 ...
1
vote
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, ...
2
votes
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{...
0
votes
0answers
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
2
votes
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,...
0
votes
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 ...
-1
votes
0answers
50 views
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 ...
0
votes
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=\...
2
votes
0answers
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 ...
1
vote
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 ...
0
votes
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
$$\phi(P) = \left\{
\begin{array}...
10
votes
0answers
59 views
If the number of points of two elliptic curves are close enough, then they are equal?
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
$
On page 23 of this paper, one can read:
For ...
-3
votes
0answers
20 views
Generalized Elliptic Curves over finite field?
I am interested about the way to classify Generalized Elliptic Curves over the finite field $\mathbb{F}_3$, but i didn't find any references.
I wish to understand the techniques used in such ...
1
vote
0answers
75 views
Find $x$ such that $x^3+ax+b$ is a square mod $p$
Let $p>3$ be a prime, and $a,b \in \Bbb F_p$ such that $4a^3+27b^2 \neq 0$.
Can I find some $x \in \Bbb F_p$ such that $x^3+ax+b$ is a square mod $p$ without using Hasse's bound?
I know that $\Bbb ...
0
votes
0answers
23 views
Computing Elliptic Curve Isogeny from knowledge of the kernel of the dual Isogeny
Let $E_0(\mathbb{K})$ and $E_1(\mathbb{K})$ be two Elliptic Curves with $char(\mathbb{K})>3$ (e.g. satisfying a Weierstrass-equation). Let $\phi:E_0\to E_1$ be a separable isogeny and $\hat\phi:E_1\...
1
vote
0answers
72 views
$\alpha$, and the $2L$ conductor in Serre's proof on $a^p+b^p+L^α c^p=0$
In Serre's paper, On the two-dimensional modular representations of $\operatorname{Gal}(\overline{Q}/Q)$, on page 25 Section 4.3, Variants of Fermat's Theorem, Serre gives a proof that
$a^p+b^p+L^α c^...
0
votes
1answer
41 views
Elliptic Curves in Cryptography [closed]
Elliptic curve cryptography is based on finding intersections of lines and elliptic curves:
$$y^2 = x^3 + ax + b ~~\text{and}~~ y = ax + b$$
It make sense when you see it on the graph, but the ...
0
votes
0answers
22 views
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
4
votes
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.
1
vote
0answers
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 ...
1
vote
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 ...
0
votes
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 =...
3
votes
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 ...
4
votes
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 ...
3
votes
0answers
91 views
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 ...
2
votes
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 $...
2
votes
0answers
48 views
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,\...
0
votes
0answers
46 views
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 ...
1
vote
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 ...
6
votes
2answers
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 ...
4
votes
0answers
89 views
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\...
1
vote
0answers
38 views
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> \...
0
votes
0answers
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 ...
3
votes
0answers
33 views
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 ...
0
votes
0answers
21 views
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 ...
0
votes
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. ...
3
votes
0answers
74 views
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 ...
7
votes
3answers
165 views
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 ...
4
votes
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? ...
0
votes
0answers
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 $d=2$:
...
2
votes
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:
...
1
vote
0answers
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^...
2
votes
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 $\pi^k <= B_1 < \pi^{k+1}$
$\quad$...
1
vote
0answers
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 =...
