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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 .

3283 questions
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Plot points of elliptic curve

I'm interested in plotting points of an elliptic curve over the real numbers. I'm looking to plot a few curves, but one like y^2 = x^3 + 7 would be an example of one. This is simple enough when x is positive, but when it's negative, I'm unsure how…
babaloo
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Equation of elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $T$. One can write a Weierstrass equation for $E$. If I define $C := E/\langle T \rangle$, what is the Weierstrass equation for $C$? Is it just the equation given at the top of page 4 in…
user60194
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point addition on elliptic curve

Bezout's theorem states that the number of common points of two curves is at most equal to the product of their degrees, and equality holds if one counts points at infinity and points with complex coordinates (or more generally, coordinates from the…
arulbero
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How Can I Know the Number of Points on an elliptic curve?

If I have the following Elliptic Curve: $$E: y^2 \equiv x^3 + 2x + 2 \pmod{17}$$ How can I calculate the number of points on this elliptic curve $E$ ? and how can I invest the following law which calculates the Number of Points on an Elliptic…
jemmy.w
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exercise 2.10 in silverman AEC

the exercise 2.10(a) in silverman AEC says $(\phi_*f)(D)=f(\phi^*D)$.(any $f\in \bar K \ (C_1),D\in Div(C_2) ) ( \phi:C_1 \rightarrow C_2 is \ a \ nonconstant \ map \ of\ smooth\ curve $ ) (Here we need the support of divisor D and the support of…
zheng
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Show that $j(E^{(d)}) = j(E)$

Let $E$ be given by $y^2 = x^3 + Ax + B$ over a field $K$ and let $d \in K^\times$. The twist of $E$ by $d$ is the elliptic curve $E^{(d)}$ given by $y^2 = x^3 + Ad^2x + Bd^3$. Show that $j(E^{(d)}) = j(E)$ I know $j(E) = 1728 \frac{4A^3}{4A^3 +…
XRBtoTheMOON
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Computing points on an elliptic curve over $\mathbb{F}_5$

If I have $E: y^2 = 2x^3 + 3$, and I want to compute all the points over $\mathbb{F}_5$ Do I simply just plug in 0-4 for y, and 0-4 for $2x^3 +3$ and then all the ones that are equal are the only points? So in this example $$(1,0), (2,2), (2,3),…
XRBtoTheMOON
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The explicit triplication formula for elliptic curves

Does someone have the triplication formula, or its reference? I know the explicit addition formula for elliptic curves, but it is too complex to get the triplication formula using it.
k.j.
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A formula for counting points on a elliptic curve over a finite field

Let $E(\mathbb{F}_p)$ is an elliptic curve, #$E(\mathbb{F}_p)=p+1-a$ and $x^2-ax+p=(x-\alpha)(x-\beta)$. Now I found the following formula here https://joeylitalien.github.io/assets/elliptic-curves.pdf (Theorem 3.3) # $…
Problemsolving
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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 $\phi:C_n\rightarrow Aut(C)$ is an action, up to $C_n$ equivariant…
M. K.
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Equivalent elliptic curves

I have this quartic which I want to change into a Weierstrass elliptic curve : $$v^2 = (p^4 - 2p^3 + 5p^2 + 8p + 4)$$ and I found a different elliptic curve to the one in the book and was told that they are the same by some change of variable. I…
shahrina ismail
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Definition of the derivative on elliptic curves modulo n

I'm trying to understand the point addition and point doubling operations on elliptic curves for the purposes of elliptic curve cryptography. I've read this Wikipedia article for the formulae:…
S. Rotos
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Pairing attack on the discrete logarithm problem

Let $p > 3$ be a prime, let $E/\mathbb{F}_p$ be a supersingular curve, let $N > \sqrt{p}$ be a prime dividing $p + 1$, and let $\mu_N$ denote the multiplicative group of $N$th roots of unity in $\overline{\mathbb{F}_p}$. (a) Prove that $\mu_N…
user495623
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Is scalar division in finite field elliptic curve not multiply by inverse of 2?

Using my understanding of finite field elliptic curve arithmetics, I've come to the conclusion that it is possible to divide a point by a scalar the same was we can multiply a point by a scalar. My method for division of 2 is simply multiply by the…
gtr32x
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Is there an analogue of Mordell-Weil theorem for other fields?

The Mordell-Weil theorem states that for an abelian variety $A$ over a number field $K$ the group of $K$-rational points of $A$ is finitely generated and abelian. What if $K$ is not a number field, e. g. Q$_p$ or a transcendental extension of…
FusRoDah
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