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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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Amount of points on an elliptic curve over $F_q$

Assume I have these two elliptic curves: \begin{align*} E:Y^2&=X^3+b_2X^2+b_4X+b_6\\ E':Y^2&=X^3+gb_2X^2+g^2b_4X+g^3b_6, \end{align*} over $\mathbb{F}_q$, where $g$ is not a square in $\mathbb{F}_q$, and $\mathbb{F}_q$ does not have characteristic…
user100659
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Are all supersingular elliptic curves defined over $\mathbb{F}_{p^2}$?

In the paper introducing supersingular isogeny Diffie-Hellman, they write Every supersingular elliptic curve in characteristic $p$ is defined over either $F_p$ or $F_{p^2}$ (see Silverman's AEC) The most I am able to conclude from Silverman's book…
PatrickStewart
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Elliptic curve division polynomials equations calculations are not matching

Weierstrass form of elliptic curves is: $y^2=x^3+Ax+B$ The first four division polynomials equations are:       $ψ_0=0$       $ψ_1=1$       $ψ_2=2y$       $ψ_3=3x^4+6 A x^2+12 B x-A^2$       $ψ_4=4 y(x^6+5 A x^4+20 B x^3-5 A^2 x^2-4 A B x-8…
user1035924
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Torsion points on elliptic curve when base field changes

There are two questions related to elliptic curve (1)Let E be an algebraic elliptic curve over k where k:algebraic closed field with characteristic zero Let K be an any extension field of k so we can view E as an elliptic curve over K Then is it…
KS M
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On an elliptic curve over a finite field, why does the line(s) connecting two points also cross the third point with suitable integer coordinates?

It's true by the field's definition, that the points could add up to get to a third point with the corresponding coordinates. But how do you prove it? For example, in the example of y^2 = x^3 -7x+10 (mod 19), (graph at…
toga
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Torsion subgroup of $y^2=x^3+4x$

I am trying to find the torsion subgroup $E(\mathbb{Q})$ of the elliptic curve $E: y^2=x^3+4x$ over $\mathbb{Q}$ which apparently is $\mathbb{Z}/4\mathbb{Z}$ according to exercise 4.9 of the book "Rational Points on Elliptic curves" by Silverman and…
Anish Ray
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Give me one non-isotrivial elliptic curve over $\mathbb{F}_2(t)$ with supersingular reduction at some place

I would like the equation of a non-isotrivial elliptic curve over the rational function field $\mathbb{F}_2(t)$ with exactly one place of supersingular reduction and I would like to know which place. I tried $$ Y^2 + tY = X^3 + tX + (t+1) \, . $$ I…
user12770
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Where does this theorem of Ogg appear?

The webpage http://www-personal.umich.edu/~asnowden/teaching/2013/679/L20.html claims that Ogg proved that the order of $[0] - [\infty]$ on $J_0(N)$ is $(N-1)/\gcd(N-1, 12).$ I cannot find any paper of Ogg that seems to prove this result, and was…
user960774
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Etymology of elliptic curve

I ask why adjective 'elliptic' for elliptic curves? I have read that for the calculus of length of arc of ellipse founds a integral of type $\displaystyle\int_0^a\sqrt{\frac{1-k^2t^2}{1-t^2}} dt$ (elliptic integral). Then call $u$ the integrand I…
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How to find 2P and 3P of an elliptic curve?

Given a point $P = (-1,6)$. $2P$ is found to be $(3,-2)$. But how? I have used the standard formula and can never get this With equation $y^2 = x^3 - 15x + 22$ $$m' = (3(x_o)^2+a)/2y_o,\\ x_1 = (m')^2 - 2x_o ,\\y_1 = y_o + m'(x_1-x_o).$$ Then $2P =…
user799790
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Number of points on elliptic curves over finite fields

I noticed that for elliptic curves of the form $y^2≡x^3+a\pmod p$ sometimes the number of points is always $p+1$ for any choice of $a$. This seems to be the case for all $p ≡ 5 \pmod 6$. Moreover, when this does not happen, i.e., for $p ≡ 1 \pmod…
RedGolpe
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Converting a point in a finite field to its real (x, y) coordinate

Let curve $A = y^2 = x^3 + 3$ and curve $B = y^2 \equiv x^3 + 3 \pmod{19}$ Let $G$ be the positive y-valued point in the curve where $x = 2$ Let $r$ be a random scalar integer, for example, $r = 5$ Compute the point $G*r$ in both curves $A$ and…
user306666
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Infinity of Right Triangle Elliptic Curve

Translate the congruent number problem into elliptic curve, we conclude that an integer $n\in\mathbb Z^+$ is area of a right triangle with $a,b,c\in\mathbb{Q}$ if and only if the corresponding elliptic curve has positive rank, viz. the curve $E:…
Chompar89
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Division of integer points on rational elliptic curves

I am reading Silverman - Difference between Weil and canonical heights, and on page 739 in Example 7.1, the author is investigating the curve $E: y^2=x^3-x+1.$ The goal is to determine $E(\mathbb{Q})$. Here is an excerpt: I do not understand the…
Douglas Molin
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Number of elliptic curves (up to isogeny) over a fintie field

For a finite field $\mathbb{F}_p$ ($p$ a prime), is there an asymptotic estimate for the number of ordinary elliptic curves over $\mathbb{F}_p$ up to isogeny? It is well-known that two ordinary elliptic curves are isogenous if and only if the…
Mathdropout
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