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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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Newbie question about multiplication maps on elliptic curves

I am quite new to elliptic curves so apologies if my terminology is totally messed up. Given an elliptic curve $E(\mathbb{F}_p):y^2 = x^3 + ax + b\text{ (mod } p)$, I am wondering if the image of multiplication map $nE = \{nP : P\in…
Gareth Ma
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Uniformizer of an elliptic curve at infinity

I have read in a solution to a problem that I was studying that $t = \frac{x}{y}$ is a uniformizer at the point $(0,1,0)$ of the curve given by $y^2z=x^3+axz^2+bz^3$, and that one can expand $x=t^{-2}+...$ and $y=t^{-3}+...$. I am not sure how one…
baltazar
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Maximal small lattice points of an elliptic curve

The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?
Silas Pike
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Adding two points in an elliptic curve gives the infinity point

I have an understanding problem that I want to geometrically solve regarding elliptic curves. It is clear how two points are added: Draw the line that passess through the points (if it is the same point, just draw the tangent) and reflect the third…
Bean Guy
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Changing variables of elliptic curves

$$y^2=x^3+Ax^2+Bx+C.$$ Change $x\longrightarrow x-A/3$, so that the new equation has the form $$y^2=x^3+ax+b.$$ Can you show step by step what operations we do in the first equation while obtaining the last equation? Why did we use the…
Tez
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Simplified Weierstrass equation for char$(K)=2$

I am working through Guide to Elliptic Curve Cryptography, and there is a section which provides the simplified Weierstrass equations and the necessary change of variables depending on the characteristic of the underlying field. One part reads If…
Cobus Theunissen
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When are two isogenies equivalent?

Suppose there are two isogenies $\phi:E \rightarrow E^\prime$ and $\psi:E \rightarrow E^\prime$ two isogenies between the same two elliptic curves. Can the isogenies be different? How can you tell?
José
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Is any imaginary quadratic field equal to $\mathrm{End}(E) \otimes \Bbb Q$ for some $E / \Bbb F_q$?

Let $K$ be an imaginary quadratic field. Is there always a finite field $k$ and an elliptic curve $E$ such that the endomorphism algebra $\mathrm{End}_{\overline k}(E) \otimes \Bbb Q$ is isomorphic to $K$ ? A refined question would be whether any…
Alphonse
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Geometric interpretation of the 9 order 3 points of Elliptic curves

Silverman in their book Rational Points on Elliptic Curves have a theorem; Theorem 2.1: Let $C$ be a non-singular cubic curve $$C : y^2 = f (x) = x^3 + ax + bx + c.$$ (c) A point $P = (x, y) \neq \mathcal{O}$ on $C$ has order three if and only if…
kelalaka
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Weirstrass Normal Form, Canonical form and j-invariant of a curve.

Given the following elliptical curve: $y^{2}=x^{3}-4x^{2}+4$ How may I write it in: Weirstrass Normal Form The Canonical Form and calculate the j-invariant There are many authors suggesting different methods: If we write it in the form:…
Sean
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Points Q of the form 2Q = P on an Elliptic Curve

I'm current reading Koblitz's "Introduction to Elliptic Curves and Modular Forms," and the author repeatedly mentions that, given a fixed point $P$, points $Q$ of the form $2Q=P$ are found by taking the lines emanating from $-P$ that are tangent to…
Johnny Apple
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How many 2-torsion points in an elliptic curve?

N torsion points have the structure ker([n]) ≅ Zn×Zn , so ker([2]) ≅ Z2×Z2 , gives us 3 2-torsion points.2-torsion points but ker([4]) ≅ Z4×Z4 ,this means we have 5 subgroup of order 4 . In each subgroup , there is a generator Pi, my question is ,…
rzxh
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What does non-identity component of an elliptic curve mean?

I have mainly concentrated on Part b) on exercise 9.13. What does it mean for a point to be not on the identity component (essentially nowhere else in the book - Silverman's - is this term mentioned). And how does exercise 9.12 (I guess part a))…
DesmondMiles
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Confusion in proof of why $\#E(\mathbb{F}_q) = \deg(F_q -1)$

If we take E to be an elliptic curve over $\mathbb{F}_q$ given by $$y^2=x^3+Ax+B$$ and $F_q$ to be the Frobenius endomorphism given by: $$F_q (x,y) = (x^q,y^q)$$ I am told that: $$\#E(\mathbb{F}_q) = deg(F_q-1)$$ In the very first line of the…
user75617
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Working on Twist Elliptic Curves?

So I was reading this online article about twist curve and stuck at the following quote, "...you are effectively working on a twist E‘ rather than E." But what does it mean exactly? For supersingular curve $E$ and its twist $E'$, their abscissas…
Taylor Huang
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