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

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1answer
20 views

Specific cartesian coordinates of an ellipse

I want to do the following: 1.) Ask user for the vertical and horizontal distances of the ellipse 2.) With this information calculate the circumference 3.) Divide the circumference by the closest ...
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0answers
67 views

Elliptic Curves

I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is ...
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0answers
37 views

What is rational point on elliptic curve over Galois field

It is clear what is a rational point on elliptic curve, when the curve is defined over real numbers. But if it is defined over Galois field, what is a rational point? If necessary, supply an example, ...
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2answers
65 views

Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
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0answers
39 views

divisors eqivalence and Picard group

I have a Lemma which I understand until a certain point. It claims that if E is an elliptic curve and $P,Q \in E$, than $\exists R\in E:P+Q\sim R+\theta$, where $\theta$ is the point at infinity. To ...
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1answer
34 views

How come $f(0) = 0$ in $\mathbb C/L$?

How come $f(0) = 0$ in $\mathbb C/L$? Does anyone know it? Your help will be appreciated. This is taken from the text "Rational Points on Elliptic Curves" by Tate and Silverman.
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0answers
73 views

Cassel's book on Elliptic Curves

Let $E/\mathbb{Q}_p$ be an elliptic curve. Then for $n \geq 1$, let $E_n(\mathbb{Q}) = \left\{P \in E(\mathbb{Q}_p) : \dfrac{x(P)}{y(P)} \in p^n \mathbb{Z}_p\right\}$. According to Cassels in Lectures ...
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0answers
46 views

the torsion subgroup of E(Q) (eliptic curves)

if $E$ is an elliptic curve over $Q$, then why $E(Q)_{\rm tor}$ is group and finite set ?
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0answers
48 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 ...
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1answer
75 views

Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in ...
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1answer
51 views

Finding the inverse of P on the generalized weierstrass equation

If P = (x, y) = ∞ is on a monic cubic polynomial, then −P is the other finite point of intersection of the curve and the vertical line through P. Show that −P = (x, −$a_{1}x$ − $a_3$ − y). (Hint: This ...
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1answer
65 views

Calculating Non-Singular Map of Elliptic Curve

I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity ...
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0answers
39 views

What is so special about Frobenius endomorphisms in elliptic curves?

What is so special of Frobenius endomorphisms in elliptic curves? Why we use it for? Does it have any severe implication?
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1answer
83 views

Weierstrass form for some equation

How to find a birational transformation that turns the equation $3(y^2-1)=2x^2(x^2-1)$ into Weierstrass form? Thanks!
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0answers
63 views

Addition a point on an elliptic curve with an integer value

Suppose $Q$ is a point in an elliptic curve such that $Q=dP$ and $d$ is an integer value, and $P$ is base point of that elliptic curve. Note $Q = dP$ means that $P+\cdots+P$ for $d$ times** and since ...
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0answers
84 views

Is Hash(bG) equal to b(Hash(G))?

Assume b is an integer, G is a basepoint in an elliptic curve, and Hash is a one-way hash function. Is Hash(bG) equal to b(Hash(G)) ? or not? Note: A hash function is any algorithm or subroutine ...
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0answers
108 views

Some basic questions about Jacobians of curves

Let $C$ be a curve defined over $\mathbb{Q}$, of positive genus. Let $J$ denote its Jacobian. I would like to ask a couple of basic (I presume) questions: 0) Why is $J$ an algebraic variety? 1) For ...
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0answers
68 views

Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?

The motivation to this question can be found in How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ We know that there is an isomorphism ...
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0answers
75 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 ...
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0answers
114 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 ...
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1answer
285 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
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1answer
93 views

Bound of point's order on elliptic curve

For a given elliptic curve over a finite field and a point $P$ on that curve, how can we bound its order (integer $k$, such that $k*P=O$).
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2answers
46 views

Tate-Shafarevich groups and Hasse principle (reference)

I'm looking for a proof of the fact that the Hasse local-global principle holds for an elliptic curve $E$ defined over $Q$ if and only if the Tate-Shafarevich group of $E$ vanishes. I just need to ...
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1answer
435 views

Doubt on class group

I started reading Class group after some one's advice ,so I got the following doubts,I would be happy if someone clarify the doubts, I understood that the class group measures the failure of the ...
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1answer
177 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 ...
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1answer
107 views

Rational Points on Elliptic Curves [closed]

Compute the group $C(F_p)$ for the curve $С : y^2=x^3 + x + 1 $ and the primes $p = 3, 7, 11$ and $13$.
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1answer
51 views

Does 0 lie on elliptic curve?

Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?
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0answers
81 views

Coverings and elliptic $\mathbb{Q}$-curves

Following http://mathoverflow.net/questions/149815/automorphisms-of-the-l-function-associated-to-an-elliptic-mathbbq-curve I consider a $Q$-curve $E/K$ defined over $K$. If I'm not mistaken, the ...
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1answer
38 views

Elliptic curves 2P, 3P

How do I compute 2P, 3P etc? ex: $y^2=x^3+4xmod7$ and I have to compute the order of (2,3)=P and my example says 2P =(0,0) 3P=(2,4) but I don't know how to get these answers?
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2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
-12
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1answer
831 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...