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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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Original motivation for pairing definitions

Today the Weil and Tate pairings are used a lot in cryptography. I'm curious, what was the original motivation of Weil and Tate for defining them? (Especially curious about Weil.)
relG
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Why does the embedding of an elliptic curve in $\Bbb{CP}^2$ look like a torus?

Consider a general elliptic curve of the form $y^2=x^3+ax+b$, where $a,b\in\Bbb{C}$. These set of notes say that embedding this curve in $\Bbb{CP}^2$ make the zero set look like a torus. I am looking for an explanation of this. Thanks!
user67803
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Why does 4 | E(K) for Montgomery curves?

Given a Montgomery curve over some finite field $K$ in the form $E/K: by^2 = x^3 + ax^2 + x$ and using $E(K)$ for the $K$-rational points. I've just read that the number of points in $E(K)$ is always divisible by 4. This is from crypto and usually…
Simon F
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Points multiplication of elliptic curve

Let an elliptic curve $ E(a,b )$ $$E(a,b ) = \{(x,y)\,|\,y^2=x^3+ax + b\}$$ Where the points of the line $xm + n$: $$P =(x_1, y_1),\, Q =(x_2, y_2),\, R =(x_3, y_3) \in E(a, b)$$ How can you calculate the product between them? $$Q\cdot P =\,…
Darío A. Gutiérrez
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Why Are There Only $2^t$ Points of Order $2$ in an Elliptic Curve

Let $E(\mathbb{Q})$ be an elliptic curve over rationals. Mordell's theorem says that $E(\mathbb{Q}) \simeq \mathbb{Z}^r \oplus (\mathbb{Z}_{p_1^{\nu{_1}}} \oplus...\oplus \mathbb{Z}_{p_s^{\nu{_s}}})$ where $p_j$ is prime. I'm reading Silverman and…
Kurome
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Change of variable for Weierstrass equation.

Suppose I have the following Weierstrass equation: $y^2-y=x^3-x$. I want to make a change of variable to get the form $y^2=x^3+ax+b$. In Milne's book, p.50, we have the following statement: "Let $E$ be an elliptic curve over $k$. Any equation of the…
usere5225321
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Hasse Invariant of an Elliptic Curve

Is there a general method to calculate the Hasse Invariant for any elliptic curve over any finite field? I have read about the Hasse Invariant on page 140 in 'The Arithmetic of Elliptic Curves' but I would like some more explanation. Thanks
Junsworth
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points on elliptic curve

I have a 4-torsion point on elliptic curve $$E : y^2=x^3-3267x+45630$$ given by $(x,y)=(15-36\sqrt{-2},27\sqrt{256\sqrt{-2} - 160})$ which I have checked that it satisfies my elliptic curve. Let $\beta=\sqrt{-2}$. So we have…
shahrina ismail
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elliptic curve point O*P (zero * point P)

I am having confusion regarding the point 0*P on an elliptic curve which I am told it is not the point (0,0). Is it the point at infinity? What does 0*P means actually?
shahrina ismail
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How to decide whether a given curve is elliptic

How to decide whether a given curve is elliptic I have the equation $y^2+2y=x^3+x^2-x-2$ (over $\mathbb C$) then which condition is true; If the RHS (polynomial in $x$) has no repeated roots, then the curve is elliptic, (I think this holds if the…
user1161
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Understanding Supersingular Elliptic Curves

I am starting out trying to understand ordinary and supersingular elliptic curves, I have read that for an elliptic curve to be supersingular over a field F, then its p-torsion subgroup must be trivial. Firstly is this the only criteria that needs…
Junsworth
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Galois action on CM elliptic curves

Let $E$ be an elliptic curve $E$ defined over a number field $K$ such that $E$ has complex multiplication by the maximal order in the ring of integers of an imaginary quadratic field $F$. Let $K^{cycl}$ be the extension of $K$ obtained by adjoining…
MPo
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Tate models and subgroups of type$(m,m)$ - re Silverman, Hindry 88

I need help in understanding a passage in a paper by Hindry and Silverman, "The Canonical Height and Integral Points on Elliptic Curves". (re. page 439) Let $ E(K) $ be an elliptic curve with multiplicative reduction at a valuation $ \nu$, and K a…
Melech
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Solve Elliptic Curve equation

Suppose you have an elliptic curve $E_{p}$: $y^{2} = x^{3} + Ax + B \mod p$ and points $P$ and $Q$ which lie on $E_{p}$. Does there always exist $n$ such that $nP=Q$? If so, how do we solve for it? Not sure how to proceed.
Jack
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Algebraic points on an elliptic curve

There is a book about rational points on elliptic curves. What about algebraic points?
quanta
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