# Tagged Questions

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

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### Lifting a real quadratic twist of an Elliptic Curve to the modular surface

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ ...
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### Isogeny of elliptic curves over $p$-adic field

If $K$ is a $p$-adic field, and $E_q$ and $E_{q'}$ are the corresponding Tate curves for $|q|,|q'|<1$, why does $E_q$ and $E_{q'}$ being isogenous imply that there are integers $A$ and $B$ such ...
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### Help with getting the formula for rational point $(x,y)$ on the $y^2 = x^3$

How to find formula for rational point $(x,y)$ on the $y^2 = x^3$ in term of rational parameter $t$. And also how would I write in form of $(x,y) =(f(t),g(t))$
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### Finding formula for rational point ($x$,$y$) on the y = $x^2$

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))
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### example of an elliptic curve without complex multiplication

What is an example of an elliptic curve $E$ without complex multiplication? This means $End(E)=\mathbb{Z}$. I know that complex elliptic curves are given by $\mathbb{C}^2/\Lambda$ for a lattice ...
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### Surjectivity of morphisms of smooth projective varieties

I have a question regarding a proof of the "surjectivity of morphisms of projective varieties" (a whole mouthfull). Though there are proofs using completeness of varieties, I am interested in an ...
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### Nagell-Lutz theorem

I'm a question about Nagell-Nutz theorem. For example, the point $P'=( \frac{31073}{2704},-\frac{5491823}{140608})$ on the curve $$y^2=x^3+8$$ does not meet the theorem Nagell-Nutz. So I can say what ...
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### J. Silverman exercise 3.12 “The arithmetic of Elliptic curves”

I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following: Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the ...
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### Weierstrass form of elliptic curve with point with order larger than 3

L.S., Studying for my exam on elliptic curves, I tried to make exercise 8.13(a) of Silvermans "The Arithmatic of Elliptic Curves", which reads: Let $E$ be an elliptic curve defined over a field $k$, ...
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### Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia ...
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### Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
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### Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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### The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested ...
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### $E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...
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### The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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### Canonical sheaf of bielliptic surface

This is an example of bielliptic surface on page 84 from Beauville's book "Complex algebraic surfaces". Let $\rho^3=1$, $\rho\neq1$ and $F_\rho=\mathbb{C}/(\mathbb{Z}+\rho\mathbb{Z})$, ...
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### How can I find $E(\mathbb F_{17})$ for the elliptic curve $E:$ $y^2=x^3+c$ where $c$ is any element in $\mathbb F_{17}^*$?

This was left as an exercise in a seminar in my college. I tried to figure it out myself, but haven't been able to make any progress thus far. I don't think it should need any non-trivial result (or ...
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### find the structure of an elliptic curve over a finite field

For the elliptic curves E1,E2,E3, and E4 defined below, determine the structure of the groups Ek(F13) by using the information given below together with a minimal amount of extra (hand) ...
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### An elliptic curve has a point of order $n$ iff $E[n]\cong\mathbb{Z}/n\times\mu_n$?

Let $E$ be an elliptic curve over some field $k$, and let $n$ be coprime to the characteristic of $k$. Then I claim that $E(k)$ has a point of order $n$ if and only if ...
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### Real elliptic curves

Is it true that the full automorphism group of a real elliptic curve is $T\rtimes\mathbb{Z}/2\mathbb{Z}$ where $T$ is either $SO_2(\mathbb{R})$ or $SO_2(\mathbb{R})\times\mathbb{Z}/2\mathbb{Z}$? If ...
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### Moduli Space of elliptic fibration

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration. I know how it is done for the generic Weierstrass ...
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### Cuspform, elliptic curves and character sums

I was trying to read through some note by Kowalski (see https://people.math.ethz.ch/~kowalski/ik-ant-exp-sums.pdf). I was interested in trying to understand the following. The author states on page ...
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### Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
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### Unramified cocycles and the Selmer group of an ellptic curve

In Silverman's book on elliptic curves, he gives a procedure to compute the Selmer group of elliptic curve $E$ relative to an isogeny $\phi:E\to E'$. I am confused about one step in the discussion. ...
In undergrad, I remember computing the group law of the nodal cubic $y^2= x^3 + x^2$ using a particularly slick parametrization. The usual parametrization of the nodal cubic is $(t^2-1, t^3-t)$, and ...