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

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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 ...
2
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1answer
46 views

elliptic curve group law

Let $C$ be an elliptic curve over a field $k \supset \mathbb{Q}$. Then given $P$ and $Q$, we can draw the line between $P$ and $Q$ (call this line $L$) and then "find the third intersection point", ...
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0answers
44 views

How to find the subgroups of the group $C(ℚ)$?

Let $C$ be a fixed 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 ...
0
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1answer
58 views

Public Key Scheme decryption. [closed]

You have been sent a message based on the following Public Key Scheme. 1) Bob chooses two large primes $\ p,q $ with $ p \equiv q \equiv 2 \pmod 3$ and computes $ n=pq. $ 2) Bob chooses integers $ e,d ...
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2answers
49 views

Elliptic Curves and “roots”

Given elliptic curve $\omega$ in $\mathbb{R}^2$ such that $y^2 = x^3 + ax + b$, how can you find how many solutions (and what they are) of $x^3+ax+b$ have a $y$ value of $0$; or as they call it, a ...
3
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1answer
68 views

seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\Spec}{\text{Spec }}$ It's a well known fact that every elliptic curve (say, over a field $k$) has a global holomorphic nowhere vanishing differential. If ...
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2answers
40 views

Does the conductor of an elliptic curve always divide the minimal discriminant?

Of course, the primes dividing the conductor are precisely those dividing the minimal discriminant. But I cannot find any source that addresses the possibility of a prime appearing to the first power ...
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1answer
49 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
4
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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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0answers
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From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
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0answers
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There are no elliptic curves over $\mathbb{F}_8$ satisfying either $\#E(\mathbb{F}_8)=7$ or $\#E(\mathbb{F}_8)=11$

This is taken from The Arithmetic of Elliptic Curves by Silverman on page 154, Q5.10(f). One way of directly solving this problem is to work out on sage all 8^5 possibilities of elliptic curves and ...
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1answer
65 views

Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero

Silverman, p. 104: Show that the polynomials $$f=x^4−b_4x^2−2b_6x−b8 \qquad \text{and}\qquad g=4x^3+b_2x^2+2b_4x+b_6$$ appearing in the duplication formula (III.2.3d) are relatively prime if ...
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1answer
28 views

$\forall p\geq 3, E:y^2=x^3+x$ satisfies $\#E(\mathbb{F}_p)=0\mod4$

The question is taking from Arithmetic of Elliptic Curves by Silverman, Q5.12 on page 154. I've managed to show the supersingular case when $p=3 \mod 4$, which was done more generally for elliptic ...
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1answer
43 views

What is the argument used to dsitinguish the cases (a) and (b)

We know from [B. Mazur, Modular curves and the Eisenstein ideal, Publ. math. IHES 47 (1977), 33-186] that if $C$ is an elliptic curve of the form ($C:y²=x³+ax+b$ with $a,b∈ℤ$), then $C(ℚ)^{tors}$ (the ...
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1answer
71 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
2
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1answer
38 views

An isogeny of elliptic curves induces a $\mathbb{Z}_l$-linear map

On page 89 of The Arithmetic of Elliptic Curves (second edition), Silverman says: Let $\phi:E_1\rightarrow E_2$ be an isogeny of elliptic curves. Then $\phi$ induces maps $\phi:E_1[l^n]\rightarrow ...
4
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1answer
122 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
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1answer
57 views

Show that there exists a constant $c$ such that for all $n \in \mathbb{Z}_+$ one has $\#\{\omega \in L\,\vert\,n \leq |\omega| \leq n + 1\} \leq cn$.

Let $L \subset \mathbb{C}$ be a lattice (i.e. $L = \{n\omega_1 + m\omega_2 \,\vert\, \omega_1, \omega_2 \in \mathbb{C},\, \omega_1 / \omega_2 \not\in \mathbb{R}, \, n,m \in \mathbb{Z}\}$). Show ...
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1answer
61 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
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1answer
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Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
3
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1answer
53 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
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2answers
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why say complex multiplication of elliptic curves is beautiful

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. Just as the title asked. ...
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2answers
63 views

ODE using Weierstrass's P function

I need a hint for the following problem. "Solve $(x')^2=x^3 − 3x^2 − 4x + 12$ with the initial with initial condition $x(0)=3$". I know I should somehow use Weierstrass's $P$ function because it ...
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2answers
76 views

Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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1answer
50 views

Prove that $E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}$.

Let $E$ be an elliptic curve over $\mathbb{C}$. We know that $E(\mathbb{C}) \cong \mathbb{C}/L$ (this is a group isomorphism) for some lattice $L \subset \mathbb{C}$. Using this fact prove that ...
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$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
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1answer
44 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
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1answer
22 views

Computing fractions Weierstrass curves and DLP problem

I am preparing for a crypto exam by making an old practise exam. I got stuck on the following assignment. I got this weierstrass curve The curve $y^2 = x^3$ is not an elliptic curve over $F_{71}$ but ...
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1answer
36 views

Finding all Points on a Edwards curve

I need to find all affine points on the Edwards curve: $x^2 + y^2 = 1 - 5x^2y^2$ over $F_{13}$ I tackle this by transforming the equation to: $y^2 = \frac{1-x^2}{1+5x^2}$ I then go from x = 0 to ...
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1answer
25 views

Addition on an Elliptic Curve and Modular Arithmetic involving fractions

I'm having a bit of an issue with addition on elliptic curves. For example, I've been given the curve $Y^2 = X^3 + 2X + 1$, working modulo 3. Now, say I want to add the point $(1,2)$ with itself. To ...
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51 views

Parametrization of this elliptic curve

What's the simplest way to parametrize the curve given by the equation $$y^2 = (x^2-a^2)^2 - b^2,$$ namely simple functions (polynomials?) $x(z)$, $y(z)$, that would satisfy the above relation. This ...
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1answer
28 views

Let E be defined over Fq and let n ≥ 1. Show that E(Fq)[n] and E(Fq)/nE(Fq) have the same order.

Let E be an Elliptic Curve defined over $F_q$ and let n ≥ 1. Show that $E(F_q)[n]$ and $E(F_q)$/$nE(F_q)$ have the same order. I feel like this is obvious. The n-th torsion group $E(F_q)[n]$ ...
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0answers
22 views

Could a hyper elliptic curve of degree 5 admit only three ordinary double points?

The definition for hyper elliptic curves is those curves which admit a ramified double cover of $P^1$. The given degree five is for some homogeneous equations of degree five that defines the curve, ...
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2answers
63 views

Reason behind standard names of coefficients in long Weierstrass equation

A long Weierstrass equation is an equation of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, ...
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3answers
88 views

The group structure of elliptic curve over $\mathbb F_p$

I want to find the group of the elliptic curve $y^2=x^3-x$ over $\mathbb F_p$ for all primes $p \le 29$. But I know only 1 fact about the structure of this group: $E(\mathbb F_p)=\mathbb Z/m \mathbb Z ...
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1answer
22 views

Explanation and validation of point adding/doubling on elliptic curves

I'd like to implement point multiplication on elliptic curves over prime fields. My problem is that I've found different definition how to do it. At adding: the second parameter of the result is not ...
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1answer
102 views

Trace of Frobenius of elliptic curve is integer

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...
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1answer
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Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
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How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
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3answers
195 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
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2answers
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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 ...
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1answer
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Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2

Problem: show that an elliptic curve over a field of char 2 has nontrivial 3-torsion points Method: I used SAGE to unwind the duplication formula for an elliptic curve given in short Weierstrass form ...
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4answers
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What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
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1answer
45 views

Mordell-Weil rank bound

Given an elliptic curve $y^2 = x(x^2 + bx + c)$ is a non-singular curve, say $c > 0$ and $b^2 - 4c > 0$. Can we show the bound on the rank $r$ in terms of $\nu(c)$ and $\nu(b^2 - 4c)$ without ...
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0answers
36 views

Constant in an inequality for a height of an elliptic curve

I am trying to find explicitly a constant $\kappa$ in an inequality for the height of an elliptic curve. Suppose the curve $E$ is defined by $y^2 = x^3 - kx$ with $k \neq 0$, the curve is defined over ...
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1answer
115 views

Why are mathematicians more interested in elliptic curves than other algebraic curves?

Why are mathematicians more interested in elliptic curves than other algebraic curves? There must be some reason that motivates mathematicians to research elliptic curves specifically.
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27 views

What is some prerequisites to start reading Knapp's elliptic curve book?

I have a pdf of Knapp's elliptic curve, it looks a lot more elemantary to other books on the same subject. However, what is some prerequisites to start reading Knapp's elliptic curve book?
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1answer
29 views

Groups of rational points invariant birational transformation

I was reading Silverman and Tate's Rational Points on Elliptical Curves, and it said something along the lines of Birational transformation preserves the structure of the groups of the points ...
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1answer
81 views

How is this an isomorphism?

$\newcommand\O{\mathcal O}$I was reading Silverman and Tate's Rational Points on Elliptical Curves. In page 21 of the same book it was written We also want to mention that there is nothing ...
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1answer
46 views

Coefficients of an elliptic curve for which the torsion group is trivial

Consider an elliptic curve in the short Weierstrass form $$ y^2 = x^3 + bx + c, $$ defined over rational numbers ($b,c$ are integers). My goal is to provide an example of congruence relations on $b$ ...