Questions tagged [elliptic-curves]
For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
1
vote
1answer
53 views
Solve the Diophantine equation $24x^4-5y^4=z^2$
I want to solve $24x^4-5y^4=z^2$ in integers not all zero, and to fix ideas, I want to find them such that x,y are coprime.
I've tried plugging in small values of $x$ and $y$ and they don't return a ...
1
vote
2answers
38 views
What does it mean to solve the equation of an ellipse as a quadratic?
An illustration required me to find out the area of the curve $$5x^2 + 6xy + 2y^2 + 7x + +6y + 6 = 0 $$
They proceeded to solve the equation as a quadratic obtaining $y_1$ and $y_2$ as the two ...
0
votes
0answers
29 views
Finding the isomorphism type of an elliptic curve
Consider the elliptic curve $E: y^2=x^3+1$ over $\mathbb{F_q}$ where $q = 15485863$ (the $1000000^{\text{th}}$ prime).
I have computed (using sage) that $P=(15065540,4435916)$ has order $5160153 = 3\...
2
votes
1answer
34 views
Number of Rational Points on $C : ax^2 + bxy + cy^2 = dz^2$ over finite field
Let $p \neq 2$ be a prime, let $a,b,c,d \in \mathbb{F}_p$ satisfy $acd \neq 0$, and let $C$ be the conic given by the homogeneous equation $$
C : ax^2 + bxy + cy^2 = dz^2. $$
a) If $b^2 \neq 4ac$...
0
votes
0answers
23 views
Construction and properties of flat family of elliptic curves
I have the following situation: let $k$ be algebraically closed of characteristic 0 (one can assume $k=\mathbf{C}$ if this simplifies the discussion) and let $\varphi : \mathfrak{X}\longrightarrow \...
1
vote
0answers
29 views
Is Elliptic Curve Discrete Logarithm Problem NP-Hard or NP-Complete
I have trouble classifying Elliptic Curve Discrete Logarithm Problem as NP-Hard or NP-Complete. Where does ECDLP belong? Any brief comprehensive answer is encouraged. Thanks.
0
votes
0answers
21 views
What is a supersingular elliptic curve over arbitrary rings
I read in Katz Mazur's book on moduli spaces of elliptic curves that an elliptic curve over an $\mathbb{F}_p$-algebra $R$ is called ordinary if its geometric points are all ordinary. Now the question ...
3
votes
0answers
37 views
Elliptic curves (Tate normal form?)
I basically have two question, the other question can be found below.
Let $E/k$ be an elliptic curve with $P\in E(k)$ a point of order $\geq 4$. Show that $E$ can be described by
\begin{equation*}
y^...
3
votes
0answers
32 views
Do we have a criterion for $j$-invariants that gives $CM$ elliptic curves?
It is well-known that every CM elliptic curve has $j$-invariants which are algebraic integers. Is there any criterion to classify all the $j$-invariant that corresponds to $CM$ elliptic curves? For ...
1
vote
1answer
25 views
Find three points of order two on elliptic curve.
Let $C$ be the cubic curve defined by $y^2z = x^3 -xz^2$ where $O = (0:1:0)$ is an inflection point. Find three points of order two in the group $(C, O, +)$.
I know that $2\cdot P = O$ if and only if ...
1
vote
1answer
24 views
Notation confusion in Hales's The Group Law for Edwards Curves
I have an issue undertanding some notation in Thomas Hales, The Group Law for Edwards Curves. At page 6, he writes:
We use the following rings: $R_0:=\mathbb{Z}[c,d]$ and $R_n := R_0[x_1,y_1,\...
0
votes
1answer
27 views
Question about using sum of quadratic residue to count points on elliptic curve in Schoof's paper
Sorry, I have a very basic question about Schoof's paper, when he is talking about taking the sum of quadratic residues to count points.
"This implies that evaluating the sum $$\sum_{x \in F_p} \...
5
votes
2answers
87 views
Calculate the number of points of an elliptic curve in medium Weierstrass form over finite field
Let $E$ be the elliptic curve over $\mathbb{F}_3$ in medium Weierstrass form $E:y^2=x^3+x^2+x+1$. How to compute the number of points $|E(\mathbb{F}_{3^k})|$? I read that there are some formulas for ...
0
votes
1answer
48 views
The group of points of an elliptic curve $E$ over a finite field $E(\mathbb{F}_q)\cong \mathbb{Z}/n\mathbb{Z} \times\mathbb{Z}/n \mathbb{Z}$
For the group of an elliptic curve $E$ over a finite field $\mathbb{F}_q$. If $E(\mathbb{F}_q)\cong \mathbb{Z}/n\mathbb{Z} \times\mathbb{Z}/n \mathbb{Z}$, then $q=n^2+1$ or $q=n^2\pm n +1$ or $q=(n\pm ...
1
vote
1answer
45 views
Rational $3$-torsion points of an elliptic curve.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with equation $y^2=x^3+(ax+b)^2$, where $a,b \in \mathbb{Q}$.
I have to prove that $0$ and $(0, \pm b)$ are the rational $3$-torsion points of $...
1
vote
1answer
38 views
Let $C$ be the cubic curve $ y^2 = x^3 + ax^2 + bx + c $ with singular point $S$. Show that $S$ must have rational coordinates.
This is exercise 3.14 of Silverman's book Rational points on elliptic curves.
Let $C$ be the cubic curve given by the equation
$$ y^2 = x^3 + ax^2 + bx + c $$
with $a,b,c \in \mathbb{Q}$. Suppose ...
0
votes
1answer
51 views
Using Velu's formulas in MAGMA
Most isogeny-based cryptographic schemes rely on constructing an isogeny having a given kernel. That is, given an elliptic curve $E$ and a subgroup $G$ of points of $E$, there is interest in ...
1
vote
1answer
38 views
Isomorphism between an elliptic curve and the additive structure of the field.
Let $C: y^2=x^3$ be a singular elliptic curve over a field K and define $C_{ns}(K)=C(K)/{0,0}$ the set of non singular points of C including $P_\infty$. I've proved that $C_{ns}$ is a group with the ...
0
votes
0answers
40 views
Not all curves of genus greater or equal to 3 are hyperelliptic, and their dimension
I'm supposed to show that for all genuses $g\geq3$, not all curves are hyperelliptic. Let me preface this by saying that I am not taking a what I think probably is a typical algebraic geometry course. ...
6
votes
1answer
146 views
Finding all rational solutions to $y^2=x^3-432$
In the book "Elliptic curves, number theory and cryptography" by Washington in section 2.5.2 it says that it is somewhat not trivial to show that the only rational solutions to the curve $y^2=x^3-432$ ...
0
votes
0answers
59 views
Parametrization of special family of tori knots
Finding the parametric equations of an (a-c)tori knot knowing that one turn has the following parametric equation:
$$\alpha(t)=\begin{pmatrix}
x=\Big(R_1+R_2\cos(t)\Big)\cos\biggr(c\arctan\Big(\dfrac{...
0
votes
2answers
38 views
Finding the Weierstrass normal form of an elliptic curve.
I am given an elliptic curve with homogeneous equation $X^3+Y^3+Z^3=0$ over a field K and I am asked to find the Weierstrass normal form.
I started by noticing that the point $(1,-1,0)$ is in the ...
1
vote
0answers
36 views
Deriving a formula for the addition of two points in an elliptic curve and finding its Weierstrass normal form.
I am given the elliptic curve with homogeneous equation $E: X^3+Y^3+Z^3=0$ over a field $K$ with a point $P_{\infty}=[1:-1:0]$. I am told that you can turn $E(K)$ into a group with identity element $...
0
votes
0answers
16 views
Misprint in Fearless Symmetry by Ash and Gross? Conditions for Elliptic Curves.
On page 104 of the paperback edition of Fearless Symmetry by Avner Ash and Robert Gross, they give one way of thinking of elliptic curves as $y^2=x^3+Ax+B$ wherein $A$ and $B$ can be any fixed ...
0
votes
0answers
17 views
How to obtain private key of Elliptic curve by factorization?
Okay i am trying to understand how the elliptic curve cryptography can be cracked by using factorization. Am not getting any clues after searching the internet can someone show me how your help will ...
1
vote
2answers
85 views
Finding all rational points in $x^2+y^2=6$.
I want to find all rational points in the circle $x^2+y^2=6$. This would be easy if I could find one rational point in the circle, however, it's very hard to guess one in this case. However, I don't ...
2
votes
1answer
56 views
irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication
Currently I am reading the book, Fermat’s Last Theorem written by Darmon, Diamond and Taylor. (You can find this pdf online http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf)
...
3
votes
1answer
113 views
Bounding rectangle of ellipse
In a stale branch of the code base I'm working on, I found an interesting algorithm to go from the SVG definition of an elliptical arc, to Bézier curves. This is a question about a small but crucial ...
0
votes
0answers
34 views
Modular Polynomial Arithmetic in Schoof's Algorithm
I've been trying to implement Schoof's Algorithm, and I understand it except for one part.
Near the bottom of page 7 of this paper is where my issue is: http://www-users.math.umn.edu/~musiker/schoof....
2
votes
1answer
86 views
The Legendre family of elliptic curves
The Legendre family of elliptic curves over the $\lambda$ line is given by the equation
$$E_{\lambda}:y^2=x(x-1)(x-\lambda),\lambda \in P^1_{\mathbb{C}},$$
which has three singular points, $\lambda=0,...
5
votes
0answers
80 views
Elliptic curves: points with nonsingular reduction. Finiteness of a quotient.
Let $E$ be an elliptic curve defined over a complete local ring K in characteristic greater than 5 (one may assume that E is given by a Weierstrass equation of the form $y^2 = x^3 + Ax + B$). Denote ...
1
vote
0answers
49 views
Parameterized Elliptic curve over Q with rank at least 1
Without explicit numbers, what tools are available to prove the rank of an Elliptic curve is at least 1?
To provide a concrete example to aid discussion, consider the curves:
$$ y^2 + y = x^3 - 18(k+...
0
votes
0answers
13 views
How to do the shortcut function in ECC when N (Private Key) is Known
When N or private key is known, we don't have to iterate through all the process just to get the final location given the two initial points. How is that shortcut function implemented given the ...
2
votes
0answers
67 views
Is this an elliptic curve $x^2+y^2=k^2(1+x^2y^2)$
Is this an elliptic curve $x^2+y^2=k^2(1+x^2 y^2)$ over rationals? If yes, how can we transform it into usual form? My friend sent me this curve.
1
vote
1answer
48 views
Methods for solving Elliptic curve over Q taking advantage of Complex Multiplication
In "An Introduction to the Theory of Numbers" by Hardy and Wright, they tantalizingly introduce a bunch of properties of elliptic curves, including the possibility of having Complex Multiplication, ...
0
votes
2answers
28 views
Commutativity of $End(E)$ for $E$ elliptic curve
If I consider an elliptic curve $E$ over a field of characteristic zero, is $End(E)$ always a commutative ring? If not, can someone give a counterexample?
5
votes
1answer
120 views
Are problems in “Arithmetica” of Diophantus all solved now?
It's well-known that Diophantus had written ”Diophantus“ which contains many problems about solving arithmetic equations. I wonder whether all of them has been solved using modern techniques, as some ...
0
votes
0answers
18 views
Modular degree in PARI/GP
For a given elliptic curve $E$ over $\mathbb{Q}$ I'm trying to find its modular degree using the math software PARI/GP.
There exists a built-in function called ellmoddegree(E), but for some reason, ...
4
votes
0answers
77 views
Determining elliptic curve analytic rank even/odd
For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if ...
2
votes
0answers
20 views
What is the area of elliptic curve determined by the metric at cotangent space?
This question may be too low-level even for this site.
For simplicity let us assume the elliptic curve to be $\mathbb{C}/\{1, \tau\}$ where $\tau=x+iy$. Let the metric on the Riemann surface to be ...
0
votes
0answers
32 views
Elliptic Curves (equation)
Is there a specific method to solve equations like $ y^2 = x^3 - n^2x $ ?
I was reading an example about $ y^2 = x^3 - 49x $ and it says that a solution is $ (-\frac{63}{16},\frac{735}{64}) $ .
How ...
0
votes
0answers
26 views
Substitute the point of infinity in a function/Elliptic Curves
I would like to ask your help explaining the following lines:
Having an elliptic curve: $E/K:y^{2}=x^{3}+3x$, where $K=\mathbb{F}_{11}$, and a function $s=\frac{x(y+x+1)}{y(x+8)}$ and it says that $s=...
1
vote
1answer
32 views
How to prove that $\text{End}_{\mathbb{F}_p}(E)$ is commutative for a given elliptic curve E?
Given a prime $p$ and considering the finite field $\mathbb{F}_p$, I need to see that $\text{End}_{\mathbb{F}_p}$(E) is commutative using orders. It is known that $\text{End}_{\mathbb{F}_p} \subseteq \...
1
vote
1answer
25 views
Why are zeroes of the elliptic curve (mod p) for integer values symmetrical about p/2
I've been reading about the elliptic curve used for key generation in bitcoin ie y^2 = x^3 + 7 (mod p) [I'm not sure how to do the congruence symbol]
To help me visualise whats going on I wrote a ...
0
votes
0answers
13 views
Proof of the decomposition of an isogeny into a separable part and a frobenius.
Lemma:
Let $\alpha:E_1\longrightarrow E_2 $ be an inseparable isogeny of elliptic
curves defined over a field $k$ of characteristic $p>0$. Then $\alpha$ can be
written in the form
$$\...
1
vote
0answers
78 views
Level Lowering obstructions
For the generalized Fermat equation:
$a^p + b^q = c^r$
with $p,q,r\ge3$ and $\gcd(a,b,c)=1$
one can construct the Frey Curve:
$y^2=x(x-a^p)(x+b^q)$
which is semi-stable for those primes $m$ ...
5
votes
0answers
103 views
Algebraic forms on an elliptic curve
On an elliptic curve defined by the equation,
$$E:y^2=x^3+a x +b$$
The algebraic form $dx/y$ is defined on the elliptic curve and it is a non-vanishing section of the (trivial) canonical bundle. From ...
1
vote
0answers
49 views
Certain Galois cohomology computation
Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
1
vote
0answers
33 views
Torsion of elliptic curves and abelian extensions
Let $L/K$ be an abelian $p$-extension of number fields and $E$ be an elliptic curve over $\Bbb Q$.
If $E[p](K)=0$, does it follow that $E[p](L)=0$ ? The converse is obviously true, but I don't have ...
0
votes
3answers
60 views
Sage code to check independence of rational points on elliptic curve
Suppose I have three rational points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ on certain elliptic curve. Then they are linearly independent if and only if the determinant of matrix $(<P_i,P_j>)_{i,j}$ is ...
