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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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Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ P_\infty$ be the point at infinity, with coordinate…
Long
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Why 1728 in $j$-invariant?

The $j$-invariant for elliptic curves has a $1728$ in it. According to Hartshorne, this is supposedly for characteristic-$2$ and $3$ reasons, despite appearances to the contrary. Indeed, it is unfathomable why it would help in char $2$ and $3$ when…
Ellptician
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Definition of the j-invariant of an elliptic curve

It seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of course, for moduli reasons, it is clear why one…
user6960
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Is the real locus of an elliptic curve the intersection of a torus with a plane?

In Lawrence Washington's book Elliptic Curves: Number Theory and Criptography I read that if $E$ is an elliptic curve defined over the real numbers $\mathbb{R}$ then the set of real points $E(\mathbb{R})$ can be obtained as the intersection of the…
Adrián Barquero
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Why is $y^2 = 1+x^4$ an elliptic curve?

I saw in a document that $y^2 = 1+x^4$ is (the affine equation of) an elliptic curve. Why is it the case? Typically, SAGE tells me it is isomorphic to $y^2 = x^3 - 4x$, which is an elliptic curve with Weierstrass equation, but I don't know how to…
Alphonse
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Division polynomials in characteristic two

Let $ E $ be an elliptic curve over a field $ K $ given by a long Weierstrass equation in Jacobian coordinates $ (x : y : z) $ $$ y^2 + a_1xyz + a_3yz^3 = x^3 + a_2x^2z^2 + a_4xz^4 + a_6z^6. $$ What should the definition of division polynomials $…
Multramate
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How does one actually compute the dual isogeny?

Given a non-constant isogeny $f : E_1 \rightarrow E_2$ of degree $n$ between elliptic curves, I'm under the impression that there always exists a unique isogeny $g : E_2 \rightarrow E_1$ satisfying $$g \circ f = [n]_{E_1}, \qquad f \circ g =…
goblin GONE
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Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?

In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at infinity is described (which in this case is an…
Gregor Botero
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Discriminant of Elliptic Curves

In the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = -16(4a^3 + 27b^2) $ or $\Delta = 4a^3 + 27b^2$ I…
Suzy
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Chapter II Example 3.3 -p.28 Silverman

I have a question about Chapter II Example 3.3 -p.28 in Silverman "Arithmetic of Elliptic Curves". I feel like I'm misreading it and would like clarification. Let $K$ be a field such that $\mathrm{char}(K)\neq 2.$ Let $e_1,e_2,e_3\in \bar{K}$ be…
Med
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The definition of an elliptic curve?

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other definition is that it is a curve $y^2=x^3+ax+b$ which is…
MBrown
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Division polynomials of elliptic curves

This is exercise 3.7 from Silvermans AEC (2nd edition). Let $E$ be a nonsingular elliptic curve over $\mathbb{C}$ given by $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ The $n^{th}$ division polynomls $\psi_{n}$ are defined using $ \psi_{1} = 1, …
Jeff Bleaney
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Is it possible to compute order of a point over Elliptic curve?

In the elliptic Curve cryptography, it is said that the order of base point should be a prime number, and order of a point $P$ is defined as $k$, where $kP = \mathcal{O}$. And to compute the order we have $P$, $\mathcal{O}$ and we need to compute…
Sudhir Kumar
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Weierstrass Form of an Elliptic Curve

Let $E/K$ an elliptic curve defined by $f(x,y)=0$. I have read that there always exists a birational transformation $(X,Y)=(X(x,y),Y(x,y))$ such that it can be written in Weierstrass form (I'm mainly interested in $K=\mathbb{R},\mathbb{C}$, so let…
Bulkilol
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admissible change of variables on elliptic curve

I was wondering whether an admissible change of variables of an elliptic curve given by a Weierstrass equation respects the group law. Let $E$ be defined over a field $K$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ If we…
Nadori
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