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Questions tagged [elliptic-curves]

For questions about elliptic curves.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$, defined over any field $K$. They form abelian groups under point addition. They are much studied in number theory, for example in cryptography and integer factorization.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=-16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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Why are zeroes of the elliptic curve (mod p) for integer values symmetrical about p/2

I've been reading about the elliptic curve used for key generation in bitcoin ie y^2 = x^3 + 7 (mod p) [I'm not sure how to do the congruence symbol] To help me visualise whats going on I wrote a script to plot the values of y * y - (x * x * x + 7)…
Derek
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When do Heegner points lead to a basis points?

For a rank 1 elliptic curve, a rational point can be obtained from Heegner points. When is this rational point a basis point? If sometimes additional work is required to obtain a basis point, is there an easy calculation to determine when this is…
PluckyBird
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Existence of rational parametrization of elliptic curves

I read somewhere that it is not possible to have rational parametrization for elliptic curves. So there is possibility of the existence of rational parametrization for a 'part' of elliptic curve or which may just miss finite number of points on the…
ersh
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Is every $\tau$ fibration an elliptic fibration?

Suppose I have a complex n-fold $X$. Let $\tau$ be a holomorphic multi-valued function from $X$ to upper half plane. $\tau$ is allowed to be singular at a codimension one locus in $X$. And let the multi-valuedness of $\tau$ be valued in…
Naruto Uzumaki
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Sum of two equal points on an elliptic curve

$E$ - elliptic curve on field $F_{2^4}$ with equetion $y^2 + y = x^3$. I need to show, that for any $P \in E$, $3P = 0$. Here is list of 16 field…
Daria
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Is there a practical use to bounding the rank of an elliptic curve over $\mathbb{Q}$ from below.

Let $E$ be an elliptic curve over $\mathbb{Q}$, then the group $E(\mathbb{Q})$ satisfies \begin{equation} E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus \mathbf T \end{equation} for some torsion group $\mathbf T$, and $r$ is the rank of $E$. I know…
Yulan van Oppen
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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 how to find the points from it. If someone could…
David Ryan
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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, then the degree is the order of the kernel which is…
ADF
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Conflicting definitions of isogeneous and the relevance of separability

The following questions seem to be related. Firstly, let $E_1$ and $E_2$ denote elliptic curves. Silverman defines that $E_1$ and $E_2$ are isogeneous if and only if there exists a basepoint preserving regular map $E_1 \rightarrow E_2$ that is…
goblin GONE
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Elliptic Curve Non-tangent Non-Vertical Line

If I plot a slightly non-vertical line that's not a tangent, it seems that it would intersect an elliptic curve at only 2 points? Is this correct? If so it throws me off my understanding the explanations that say that you always intersect 3 points…
user2297550
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Order of point on Elliptic curve

I am trying to work out an example problem in which I have to find order of $P(1/2,1/2)$ on the elliptic curve $ y^2 = x^3 + x/4 $ . So Far I have done the following: 1) Given equation of the curve, found out the equation of tangent at P (it is $y …
Meseeks
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Elliptic curves with form$ 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 rank 0 ?
user48959
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Clarification on local global criteria

I came across this sentence in one of the material I was reading this below : A major result is the Hasse-Minkowski Principle, which implies that a curve C has a point over $\mathbb{Q}$ iff it has a point over $\mathbb{R}$ and over every local …
shahrina ismail
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Elliptic curve point at infinity

The definition of the elliptic curve: An elliptic curve over K is a curve given in $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$ over K with discriminant not 0 and the point at infinity $O$. But if I use a birational transformation between two elliptic…
matzzzz
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Number of affine points on an elliptic curve

I'm given an elliptic curve $y^2=x^3+ax+b \in \mathbb{Z}_p[x]$ (with numbers $a,b,p$ not greater than $10^6$). I would like to find, using the naive approach, the number of affine points on the curve without the point at infinity . My approach is…
Artem
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