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Results tagged with elliptic-curves
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user 478525
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
3
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1
answer
303
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Non-modular elliptic curves
Do we have any examples of non-modular elliptic curves over number fields $K \neq \mathbb{Q}$?
In particular, I came across a paper by Freitas, Le Hung, and Siksek, "Elliptic curves over real quadrati …
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5
votes
2
answers
306
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Model of an elliptic curve with p-torsion
Suppose I have an elliptic curve $E$ defined over a number field $K$.
I know that if it has
a $2$ $K$-torsion, it has a model of the form:
$E: Y^2=X^3+aX^2+bX$
a $3$ $K$-torsion, it has a model of …
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2
votes
0
answers
70
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Detect all isogenies of an elliptic curve over a given number field
Given $K$ a number field and $E/K$ an elliptic curve, is there an algorithm which gives all the elliptic curves $F/K$ isogenous to $E$ (up to isomorphism)?
Or is there a bound on how many $F/K$ are …
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4
votes
1
answer
553
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Are Frey elliptic curves semi-stable?
Are all Frey elliptic curves semi-stable? If so, where exactly is this needed in the modularity approach, now that we know modularity for all rational elliptic curves?
Thank you!
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1
vote
0
answers
86
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Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion mo …
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1
vote
0
answers
170
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Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius a...
I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description.
A point $P \in X_0(p)$ can be viewed as $P:=(E, …
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2
votes
1
answer
250
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Twist of the Tate Curve
Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate curve …
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2
votes
1
answer
350
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Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan
I am trying to understand the following. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication given by the ring of integers $\mathcal{O}_K$. We are given a fixed rational prime $p$ whic …
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0
votes
0
answers
86
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Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\ …
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3
votes
1
answer
239
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Computing explicit isogenies between elliptic curves over different kinds of fields
I have some questions about isogenies of elliptic curves in two settings:
1. Elliptic curves defined over the rationals.
1.1. Given two elliptic curves $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ we can decide …
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