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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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Show that if re $\tau$ is 0 or -1/2, then $g_2(\tau)$ and $g_3(\tau)$ are both real

The question is from Washington's Elliptic Curves: Number Theory and Cryptography, Question 9.5. The previous parts (which I have solved) are (i) $\overline{j(\tau)}=j(-\overline{\tau})$, (ii) if $\tau$ is in fundamental domain $\mathcal{F}$, then…
Kenny Wong
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What is the point $\{∞\}$?

The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e., $C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$. 1) What is the points $\{∞\}$? 2) If we ignor the…
Safwane
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Find the coordinates of three distinct points on the elliptic curve

I am studying elliptic curves. I have an operation that I want to resolve. $y^2 = x^3 + 8x + 3$ on the field $\mathbb {F}_{23}$. And I try to find the coordinates of three distinct points on the curve that are not the infinite points. Can someone help…
user2388887
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Given a point R on an Elliptic curve find two distinct points P and Q so that (a) P+Q=R and (b) the y coordinate is the same for P, Q and R

Point addition on elliptic curves allows us to determine a point R given two distinct points P and Q so that P + Q = R. There are various ways to calculate point addition depending on the type of curve. A brief explanation of point addition is given…
Robert
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Duplication Formula, Elliptic Curves

I am trying to add points on an elliptic curve and failing. (Using formulas from William and Tate Rational Points of Elliptic Curves.) $$ C: \; y^2 = x^3 + x^2 +x + 3 \quad \text{over the field} \quad \mathbb{F}_{103}. $$ $P = (7,14)$ $a=1, b=1,…
Peter_Pan
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Are generators of elliptic curves unique?

Example of an elliptic curve of rank 2 with torsion $Z/3Z$: $$y^2 + x*y + y = x^3 - 75*x + 242$$ Torsion points $T_1, T_2,T_3$ are: ${(0 : 1 : 0), (4 : -6 : 1), (4 : 1 : 1)}$ Generator points $Q_1,Q_2$ (according Sage) are:${(-10 : 8 : 1), (-3 : 22…
azerbajdzan
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Transforming a cubic equation from a "diagonal" form to a Short Weierstrass form

I'm working on a HW problem from one of MITs open course (Intro to Arith Geom). The problem is to show that the cubic curve $x^3+y^3+z^3=0$ can be transformed, with a suitable change of variables, to the form $y^2z=x^3+Axz^2+Bz^3$, where the…
Jesse
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Elliptic curve addition algorithm for (0,0)

It seems to me that the Elliptic Curve Addition Algorithm is undefined for the case $P+P$ if $P=(0,0)$. Are there any alternative computation methods for this case?
jvdh
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Elliptic curve point comparison

If Alice uses point doubling over a generator point P to create two new points and then hands them over to Bob, can Bob tell which of the two points is the greater point without having to bruteforce all possible integer multiples of P? Assume Bob…
user306666
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Rank of Elliptic Curve, $Y^2=x^3+px$ where $p$ is prime is either $0,1,2$

I am following book "Rational Points on Elliptic Curves" by Silverman-Tate(basic version not the "The Arithmetic of Elliptic Curves" by Silverman-Tate) And I am trying to solve for cubic curve, $y^2=x^3+px$ where $p$ is prime 1) Rank of the curve is…
mani
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Find isogeny between two given points

Let $P$ be a point on an elliptic curve $E$ and let $Q = \phi(P)$, where $\phi: E \to E'$ is an isogeny of degree $d$. Given $E, E', P, Q$ and $d$, is it possible to find an isogeny $\phi': E \to E'$, not necessarily equal to $\phi$, such that…
Andrea
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good/bad reduction modulo $p$ in Rational Points on Elliptic Curves vs. Arithmetic of Elliptic Curves

I am trying to understand what an elliptic curve mod $\pi$ vs mod $p$ is. Basically I am confused about the treatment given in Silverman's two books. The definition for mod $p$ in Rational Points of Elliptic Curves is basically sending the group of…
quietkid
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$\deg (\alpha) = \#\operatorname{Ker}(\alpha)$ if $\alpha$ is a separable endomorphism of an elliptic curve

In "Elliptic Curves: Number Theory and Cryptography" we see in the proof of proposition 2.21 that for the separable endomorphism $\alpha : E(\overline{K}) \rightarrow E(\overline{K})$ for every point $(a,b)$ with $(a,b) \neq \infty$ and $a,b \neq 0$…
NightRain23
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Elliptic Curve scalar multiplication on $\mathbb{R}$

I have an elliptic curve $ y^2=x^3+109x^2+224x$ and a point $P(-100;260)$ on it. And I need to find point $2P$. I took a formulas $$x_2=\left(\frac{ax_1-b}{y_1}\right)^2 -a+x_1$$ and $$y_2=-y_1+\frac{ax_1-b}{y_1}(x_1-x_2)$$ put into this formulas…
aid78
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Elliptic curves: twists and homogeneous spaces

I have a slight confusion with these 2 concepts. It is my understanding that twists of an elliptic curve $E/k$ are elliptic curves $E'/k$ with $j(E)=j(E')$. Then in Chapter X of The Arithmetic of Elliptic Curves, Silverman introduces principal…
dmontana
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