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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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Finding J-invariant of Legendre form of Elliptic Curve

PROBLEM: Put the Legendre equation $y^2 = x(x − 1)(x − λ)$ into Weierstrass form and use this to show that the j-invariant is j = $2^8\frac{(λ2 − λ + 1)^3}{λ^2(λ − 1)^2}$ . Recall: Weierstrass equation form: E: y^2 = x^3 + Ax +B and J(E) =…
Adam Staples
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Violating assertion in Cohen's instructions for Weierstrass reduction

I am trying to follow case 2 of the procedure given in Cohen: for the cubic $f(x,y,z) = x^3 + 3 y^3 - 11 z^3$ using the rational point $P_0 = (2 : 1 : 1)$. The tangent at this point is $y = - \tfrac{12}{9}(x-2) + 1$ and it intersects the curve…
user16697
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Why are we not allowed to rescale the variables of the equation of an elliptic curve independently one from the other?

It seems that in whatever proof of the theorem that an elliptic curve can be put in Weierstrass form that you look at, the next step after getting an equation: $$\alpha Y^2Z + a_1XY Z + a_3Y Z^2= \beta X^3+ a_2X^2Z + a_4XZ^2+ a_6Z^3$$ is to multiply…
Rodrigo
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Isogenies between curves

Is it always possible to find an isogeny from a hyperelliptic curve of genus 4, to a 'normal' elliptic curve (genus 1), or a product of elliptic curves? Are such isogenies easy to compute?
ted.k
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Solutions of elliptic curve in finite field

If I take the following elliptic formula over a finite field of size $17$: $$y^2 = x^3 + 2x + 3$$ The solutions for $x = 2$ would be $7$ and $10$. Because $7^2=49$ and $49 \equiv 15 \bmod 17$ $10^2=100$ and $100 \equiv 15 \bmod 17$ My question is…
Niels
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Bound of point's order on elliptic curve

For a given elliptic curve over a finite field and a point $P$ on that curve, how can we bound its order (integer $k$, such that $k*P=O$).
ted.k
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Formal group of an elliptic curve, from Silverman's the arithmetic of an elliptic curves

In the beginning of page 120, to establish the formal group law for an elliptic curve, the book adds 2 points $(z_1,w_1)$ and $(z_2,w_2)$, where $w_1 = w(z_1), w_2 = w(z_2)$ using the group law. It makes explicit calculation, and argues that the the…
Long
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Determining elliptic curve's parameters from addition procedure

Given a procedure that adds two points on an unknown elliptic curve, is it possible to determine curve's parameters, treating this procedure as a black box? We are given two points on this curve $P$ and $Q$ ($Q=k*P$, for some $k$), and a function…
ted.k
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Transform Weierstrass equation into cubic

How can I transform an elliptic curve over the real numbers in Weierstrass form $y^2=x^3+ax+b$ into a cubic of the form $y^2=x(x-c)(x-d)$?
anna90
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About How to Elliptic Curve Equation and Discriminant

I am study public key cryptology and interested in elliptic curve cryplogical algorithms. I have some problem about elliptic curve equation. First I can't find inter process and transformation steps from general curve equation to elliptic curve…
Aziz YÜCELEN
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divisibility of isogenies by constant map

Let $E_1, E_2$ be elliptic curves and $\phi:E_1 \rightarrow E_2$ be a (separable) isogeny such that $\phi(E_1[m])=\{O_{E_2}\}$. Then $\phi$ is divisible by the multiplication by $m$-map $[m]$, i.e., there is an isogeny such that $\phi =…
Jason Dil
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Schoof's algorithm single or multiple output?

Shouldn't Schoof's algorithm be taking generator point as an input ? All implementations I tried (including Sage) need only the coefficients of the Weierstrass equation and a prime. No mention of the generator point at all. For elliptic curve y2 =…
user1035924
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Another form of Schoof's algorithm

All searches I did for Schoof's algorithm to count points on elliptic curves over finite fields show the same as in Wikipedia https://en.wikipedia.org/wiki/Schoof%27s_algorithm This algorithm says under section "Computation modulo primes" that "The…
user1035924
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Elliptic Curve Group

For $E_{11}(1, 7)$, compute the multiple of G from 2G through 13G. $G = (3, 2)$ I have calculated the group with 15 points (including the point at infinity O) $y^2 \pmod{11}$ y 0 1 2 3 4 5 6 7 8 9 10 Result 0 1 4 9 5 3 3 5 9 4 1 $y^2 =…
anonymous
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elliptic curve $Y^2-X^3+ p X$ with p equiv 7 mod 16 question

The elliptic curve $C_1:Y^2=X^3+pX$ has rank $0$ where is $p$ is a prime equivalent to $7$ mod $16$. This is exercise 3.8b) in the Silverman-Tate book “Rational Points on Elliptic Curves”.. In the book by Silverman “The Arithmetic of Elliptic…
Jeff Margrave
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