# Questions tagged [elliptic-curves]

Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

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### Field over which CM endomorphism is defined

Let $E$ be an elliptic curve with coefficients over some number field $K$. Is it true that if $E$ has complex multiplication by $\mathbb{Q}[\sqrt{-D}]$, then any endomorphism $\phi: E \rightarrow E$ ...
1 vote
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### Differential of Elliptic curve with bad reduction

Consider the elliptic curve defined by the Weierstrass equation $$E:y^2 = x^3+2$$ This defines a minimal Weierstrass equation for $p=2$ over $\mathbb{Q}_{2}$. It moreover has bad reduction at $p=2$. ...
1 vote
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### Fermat curve in Weierstrass form.

In this post (Substitutions that transform Fermat Equations to Elliptic Curves) it is proved that there exists a change of variables that trasform Fermat's curve $X^3+Y^3+Z^3=0$ into $y^2=x^3-432$, ...
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### Understanding “Notes on Elliptic Curves I”

Attempting to read the above paper by Birch and Swinnerton-Dyer. For reference, I’ve read to chapter 12 in Lawrence Washingtons book on elliptic curves but cannot understand Lemma 1 and Lemma 2 in the ...
1 vote
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### Given a fixed hash value (h) and private key (p), is it possible to find a nonce (k) that fits the following equation in ECC mathematics?

Assuming secp256k1 curve and ECDSA parameters, I'm trying to see if there's a way to solve for $k$, where: $k = {-h\over r} -p$, where $k$ is the ECDSA nonce, $p$ is the private key, $h$ is the hash ...
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### Why is set of elliptic curves Zariski open in parametric space $\mathbb{P}^9$?

I'm having trouble seeing why set of elliptic curves is Zariski open in the parametric space $\mathbb{P}^9$ of all cubics. I will be grateful to receive any help on this.
1 vote
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1 vote
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### Characteristic polynomial of endomorphism of the Tate module of an elliptic curve.

In Milne's book Elliptic Curves, he states (Corollary 3.23) that for any endomorphism $\alpha$ of $E$, we have the following facts about the induced endomorphism $\alpha$ of the Tate module $T_\ell(E)$...
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### Question on elliptic curve of Weierstrass form $y^2 = x^3+ax+b$: Any class there?

I want to present a brief question. I'm curious whether there is any class of Weierstrass form $y^2 = x^3+ax+b$ that we can assign them as rank $0$ by some particular property. In other words, is ...
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### Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

This is intended to be an example of an elliptic integral but I'm not sure how to go about showing it. I'm not sure which identities to start with, any tips would be greatly appreciated!
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### prime index coefficients of $L$-functions of Elliptic Curves over $\mathbb{Q}$ and analytical rank [closed]
I have searched without success if there was any known relationships or conjectures between the rank and the prime index coefficients of the $L$-functions of a modular elliptic curves. Anyone has a ...