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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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Help needed in determining the singularity

Can someone teach me how to determine the singularity of algebraic curve $y^2 =x^3+x^2$. I'll be really grateful and thanks in advance
etet112
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help to parametrize $y^2 = x^3 -x^2$

I appreciate if someone could help me to parametrize this equation $y^2 = x^3 -x^2$. Thanks in advance. I used maple to find the solution as $(x,y) = ((t^2-1),(t(t^2-1))$
etet112
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What is more amazing?

$S$ denotes the set of rational points of any curve in the plane. What is more amazing between a) and b)? a) $S$ is dense in the curve $y^2=x^3-2^4\cdot3^3\cdot7^2$ b) $S=\emptyset$ in the curve $y^2=x^3-2^4\cdot3^3\cdot5^2$ N.B.- Both, a) and b)…
Piquito
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Reflect point in Group law on elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve. When we add two points on an elliptic curve, we take the line joining them, take the third intersection point and then reflect the point and use that as the sum (at least in the generic case). Why do we…
user283918
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Example of Constant Morphism

I am reading Basic Theory of Elliptic Curves, there I came about a statement saying : A Morphism of curves is either Surjective or Constant. While studying Isogenes I came across examples of Surjective morphism but I am still wondering for an…
xyz
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Addition of x-coordinate on elliptic curve given by Möbius Transformation

Consider the elliptic curve $y^2=(x-\alpha)(x^2+ax+b)=x^3+(a-\alpha)x^2+(b-a\alpha)x-\alpha b$ over the field $K$ with $\text{char}\ K\not= 2$. The questions I am doing asks for a formula for the $x$-coordinate of $(x_1,y_1)+(\alpha ,0)$ in the form…
Andrew
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can someone explain Nagell-Lutz theorem

(elliptic curve $y^2 = x^3 + ax^2 + bx + c$) Nagell-Lutz theorem: If $p(x, y)$ is finite order on a given integer coefficient elliptic curve satisfy: (1) x and y are integer (2) y = 0 or y | D (D is discriminant) (3) there is finite such points here…
bsdshell
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Integer points belonging to two distinct elliptic curves.

Two different circles can have an integer point in common (for example, $P=(1,1)$ belongs to both $x^2+y^2-2=0$ and $x^2+y^2-4(x+y)+6=0$) but any pair of distinct elliptic curves on the class defined over $\mathbb {Q}$ by $X^3+Y^3=A$ where $A$ is a…
Piquito
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Topics in elliptic curves over finite fields

I have to write a paper on elliptic curves over finite fields and I was wondering if anyone had any ideas of some interesting directions to take this? Like what are some subtopics within this general topic? Not sure if I tagged this correctly for my…
Math Major
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Birational transformation of Elliptic curves?

Let $F:V\to W$ be a birational transformation of elliptic curves; let $g$ be a generator of $V$. Is necessarily $F(g)$, a generator of $W$?
Piquito
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How to show rational points of finite order on an elliptic cure are closed under addition

I would like to show that rational points of finite order on an elliptic curve are closed under addition. If $P_1$ and $P_2$ are rational (actually integral) points of finite order, say $nP_1= O$ and $mP_2=O$, I would like to say: $$O=nmP_1 +nmP_2…
user12802
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Point of elliptic curve

How can we calculate the multiple of a point of an elliptic curve? For example having the elliptic curve $y^2=x^3+x^2-25x+39$ over $\mathbb{Q}$ and the point $P=(21, 96)$. To find the point $6P$ is the only way to calculate: the point $2P=P+P$,…
user175343
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Elliptic curves find points with rational coordinates

The elliptic curve $y^2=x^3+3x+4$ has points O,(-1,0) and (0,2). Find five more points with rational coordinates. The answer to this example gives: (0,-2) (5,-12) (5,12) (71/25,744/125) and (71/25,-744/125) It seems to me that by changing the y…
Math Major
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How to compute $E[24]$ for $E: y^2=x^3-15x+22$

If I have an Elliptic curve $E: y^2=x^3-15x+22$ over $\mathbb{Q}$ with CM from the imaginary quadratic field $\mathbb{Q}(\sqrt{-3})$ then how do I compute the $24$-torsion subgroup $E[24]$ over $\overline{\mathbb{Q}}$? I know how to compute the…
Anish Ray
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Why complex multiplication between torus is holomorphic?

I am reading Rational Points on Elliptic Curves by Silverman and Tate. At page233, I'm having trouble. So I need your help. Let $L＝｛aw1+bw2:a,b∈\Bbb Z｝$ Then complex multiplication $φ：C（\Bbb C）→C（\Bbb C）$　induces a map $f:\Bbb C /L→\Bbb C /L$. The…
user695664
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