For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ ...
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1answer
72 views

Automorphisms of elliptic curve

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?
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Prove the nonexistence of $p$-torsion for $p > 3$ in $E:y^2 = x^3 + ax$ for prime $a \geq 2$.

$$\Large\textbf{Problem}$$ Let $E$ be an elliptic curve defined by $y^2 = x^3 + ax$ where $a \in \mathbb{Z}$ is fourth-power free. Then \begin{aligned} E(\mathbb{Q})^{\text{tor}} = \left\{ ...
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1answer
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Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
4
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1answer
141 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
2
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1answer
74 views

Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
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1answer
72 views

Trivial torsion subgroup

I am just wondering, suppose we have a curve $y^2 = x^3+ax + b$ defined over $\mathbb{Q}$ and suppose for simplicity $a,b \in \mathbb{Z}$. Can we say something about the torsion subgroup with the only ...
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0answers
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Elliptic curves in $\Bbb P^3$

How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by ...
4
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2answers
69 views

Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...
4
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1answer
106 views

Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
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0answers
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calculate canonical height ellyptic curve

Hello everyone this is my first post here and this is probably an easy question, but i am not a math geek.... I need to calculate the canonical height af a Point on an ellyptic curve in python.... I ...
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2answers
116 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
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1answer
316 views

Finding points on an elliptic curve

I have an elliptic curve $$x^3+17x+5 \mod 59$$ $P = (4,14)$ is given and I need to find point $8P$. to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = ...
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1answer
161 views

Order of a point on an Elliptic Curve

I am currently struggling with the determination the order of a point on an elliptic curve. We had to do the following exercise: $C = V(y^2+x^3-1)$ and $P = (0,1)$. Now Wikipedia told me that I can ...
3
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1answer
161 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
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1answer
82 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
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Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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0answers
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Rationality of divisors at infinity

In an attempt to clarify to myself some terminology (ant the scope of the Riemann-Roch theorem), I would like to ask for examples of genus $1$ curves of the form $$C : y^2 = ax^4 + b$$ where $a, b \in ...
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1answer
70 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
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1answer
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Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...
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1answer
84 views

Odd torsion of elliptic curves are isomorphic

$C: Y^2=X(X^2+aX+b)$ $D: Y^2=X(X^2+a_1X+b_1)$ where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$ Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which ...
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2answers
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Geometric picture of 3-torsion points on an elliptic curve

I'm faced with what seems a paradox. If we have an elliptic curve $E/\Bbb C$ in Weierstrass form so that $\mathcal O_E$ is at infinity, then the addition law is quite easy to picture geometrically. In ...
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1answer
83 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
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2answers
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There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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1answer
33 views

Why does an isogeny not ramify?

The following argument is, I believe, based on the premise that an isogeny (or a morphism of curves that is a group homomorphism) doesn't ramify: Considering the multiplication by $n$ map $[n]$ on a ...
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0answers
21 views

Local coordinate of a (hyper)elliptic curve at infinity

I would like to ask for some help to clarify the following: In (9) of page 7 of http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf one finds stated that $(\mu, \lambda) \mapsto ...
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1answer
38 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...
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1answer
75 views

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 ...
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1answer
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Conditions such that $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})\not\equiv 0 \bmod{p}$

Let $E/\mathbb{Q}$ be a semistable elliptic curve. Let $\ell$ be a prime of multiplicative reduction and consider $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})$. Given a prime $p \neq \ell$, are there any ...
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1answer
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Challenge from Fermat

Fermat challenged Frenicle with finding a pythagorean triple (a,b,c) where $(a-b)^2-2b^2$ is itself a square. By making the substitution $a=m^2-n^2$, $b=2mn$, and $c=m^2+n^2$ into $(a-b)^2-2b^2=d^2$ ...
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1answer
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Are all elliptic curves from $w^3 = \text{cubic}(z)$ isomorphic?

I've been playing around with Riemann surfaces of cubics, and it seems to me that all surfaces obtained as coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a ...
5
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1answer
117 views

Discriminants and Weierstrass form of elliptic curves

I'm confused by what appears to be contradictory information. In this post, the claim is made that "Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^{2}= x^{3}+ax+b$ where ...
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2answers
40 views

Is torsion subgroup of elliptic curve birationally invariant?

It's probably a very basic question: Having two birationally equivalent elliptic curves over $\mathbb{Q}$ - is the torsion subgroup unchanged under the birational equivalence?
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Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup

Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup. We've been told that for this problem, we are not allowed to use Mazur's Theorem. ...
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1answer
51 views

Show that the curve $2Y^2 = X^4 - 17$ has no points in $\mathbb{Q}$

There is a hint to show that if there were points in $\mathbb{Q}$, then there would exist $r,s, t \in \mathbb{Z}$ with $\gcd(r, t) = 1$ such that $2s^2 = t^4 - 17r^4$ , and then show that any prime ...
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0answers
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Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
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1answer
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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= ...
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1answer
76 views

Show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$

I've been asked to show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$ - I've managed to show that it is birationally equivalent to the curve $Y^2 = 2X^4 - 34$ (as suggested in the ...
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1answer
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Birational Transformations question

so I'm wondering, is there a birational transformation one can make to the equation $Y^2 = X^m + f_{m-1}X^{m-1} + ... + f_0$, where all $f_i \in \mathbb{Q}$ so it is of the form $Y^2 = X^m + ...
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4answers
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Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
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1answer
86 views

E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, …

Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove $ $#E($F_{2^n})$$ = \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ...
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1answer
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Rational Points on Elliptic Curves [closed]

Compute the group $C(F_p)$ for the curve $С : y^2=x^3 + x + 1 $ and the primes $p = 3, 7, 11$ and $13$.
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154 views

Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb ...
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1answer
96 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
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1answer
74 views

Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in ...
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2answers
169 views

Question about kernel of reduction mapping of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has good (nonsingular) reduction $\tilde{E}$ modulo some prime $p$. Denote their groups $E(\mathbb{Q})$ and $\tilde{E}(\mathbb{F}_p)$ ...
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1answer
150 views

Elliptic Curves Without Geometry

Unfortunately geometry terrifies me, so I was hoping to understand the basic theory of elliptic curves algebraically (via their function fields). Let F be a transcendence degree 1 extension of ...
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1answer
170 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
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1answer
68 views

Semistable Elliptic Curves

For a general representation $\rho: G_{\mathbb{Q}} \rightarrow \operatorname{GL}(V)$, where $V$ is a two dimensional $\overline{\mathbb{F}}_p$ vector space, the level $N(\rho)$ in Serre's conjecture ...
2
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1answer
46 views

Minimal Discriminant of An Elliptic Curve

I want to determine the minimal discriminant of $$ y^2 + xy = x^3-x^2-50x+111 $$ as an elliptic curve over the rationals. I managed to reduce it to the form $y^2=x^3+Ax+B$ where $A,B$ are rational, ...