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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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How can I prove that $6y^2 = x(x + 1)(2x + 1)$ is a Weierstrass equation of an elliptic curve over rational numbers

How can I prove that $6y^2 = x(x + 1)(2x + 1)$ is a Weierstrass equation of an elliptic curve over rational numbers ? How can I also find how many 2-torsion points has ?
Manthos Prokos
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How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the subgroup of elements of finite order in $C(ℚ)$).…
Safwane
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Elliptic curve question

Let $P$ be a point on an elliptic curve over $\mathbb{R}$. Give a geometric condition that is equivalent to P being a point of order (a) $2$ , (b) $3 $ , (c) $ 4$ . Could someone explain this to me in dummy terms? :) I have a test next week and…
Swayy
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Silverman AEC: exercise 3.30

Supposing A is finite abelian with #A = $N^r$ and for every $d | n$ we know #$A[D] = D^r$, with $A[D]$ the subgroup with all elements of order $D$. How do you prove that $A \simeq Z_N^r$ ? I undestand that the structure theorem for finite abelian…
NightRain23
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Independence of points on Elliptic curve

Let $P_1(x_1,y_1),P_2(x_2,y_2)...P_n(x_n,y_n)$ be $n$ rational points on given Elliptic curve. How do we prove they are independent? Are there any theorems/results/algorithms/softwares to prove their independence?
ersh
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Property of elliptic curves with a torsion point

Let $E/\mathbb{Q}$ be an elliptic curve with a $p$-torsion point. Does this imply that $E/pE$ is isomorphic to $E$? If not, are there any conditions I can assume such that this is true?
qio18
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Why can't you divide by 0 in an Edwards Curve?

The equation of an Edwards curve is $x^2 + y^2 = 1 + d x^2 y^2 \,$ The addition formula is $(x_1,y_1) + (x_2,y_2) = \left( \frac{x_1 y_2 + x_2 y_1}{1 + dx_1 x_2 y_1 y_2}, \frac{y_1 y_2 - x_1 x_2}{1 - dx_1 x_2 y_1 y_2} \right) \,$ What is the…
InterC
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Elliptic curve formulas for point addition

How can one derive the underlined formulas that are used in point addition of elliptic curves? The text is taken from https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication. I assume that $\lambda$ is the slope and that then…
torgny
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Discriminant of an elliptic curve

I have found different discriminants for general Weierstrass elliptic curves: $y^2 = x^3 + ax + b$ For example, WolframAlpha state it is $-16(4a^3 + 27b^2)$ on their site http://mathworld.wolfram.com/EllipticDiscriminant.html However, in books I…
Alice
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Better parametrization for computing group law of nodal cubic?

In undergrad, I remember computing the group law of the nodal cubic $y^2= x^3 + x^2$ using a particularly slick parametrization. The usual parametrization of the nodal cubic is $(t^2-1, t^3-t)$, and if you use this parametrization, and solve for…
Sachi
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Global minimal model of elliptic curve over $\mathbb{Q}$

I am basically trying to solve the cannonball problem using elliptic curves. In other words I have to show that the only integer points on the "elliptic curve" $6y^2 = 2x^3 + 3x^2 + x$ are $(0,0), (1,\pm 1), (24,\pm 70)$. Now my plan is to find the…
fretty
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Isogenies between elliptic curves with specified torsion groups

For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert normal form or Tate normal form. I have been…
Jesper Petersen
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Question on why short Weierstrass can't be used for curves with char=2

An elliptic curve given by $E: y^2=x^3+ax+b$ with $a,b \in K$ and $Δ(E)=-16(4a^3+27b^2) \neq 0$ is adequate for elliptic curves with $char\neq2,3$ Because of the factor -16 in the definition of $Δ(E)$, according to this definition there are no…
Math Major
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Formal Group for the Elliptic Curve $Y^2=X^3+AX$

I'm trying to solve the following problem without resorting to a direct calculation: Let $E : Y^2 = X^3 + AX$, where $A \in \mathbb{Z}$ and $A \ne 0$. Let $F(X, Y )$ be the formal group associated to $E$ and let $F(X, Y ) = \sum F_n(X, Y )$, where…
Dominic
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How to know whether an elliptic curve has a low-degree isogeny?

Given an elliptic curve with a Weierstrass equation, is there any easy way to see whether it has got an isogeny of low degree?
Alan Lee
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