# Tag Info

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### Can the Selmer curve be viewed as an elliptic curve?

Yes, for example you can adjoin an algebraic element $\alpha$ which is a root of $3X^3 + 4$ to $\mathbb Q$ to get a number field $k$, upon which this curve obtains a rational point $P = [\alpha:1:0]$. ...
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### Torsion subgroup of elliptic curve

We can apply the strategy used in this post, which uses the following proposition (Proposition VII.3.1(b) of Silverman's The Arithmetic of Elliptic Curves.) Here $K$ is a local field with residue ...

### The lattice $\displaystyle \Lambda$ associated to the pendulum equation is rectangular

For the first part, let $$v = -u-E/3$$ so that $$v'^2 =v^3-v(1+E^2/3)+2E/3-2 E^3/27$$ which corresponds to the elliptic curve C:y^2=x^3-Ax-B, \qquad A = 1+\frac{E^2}3,\quad B=\frac{2E}3- \frac{2E^3}...
1 vote
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1 vote
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### Characterization of Isogenous Elliptic Curves over Finite Fields

Yes, this is true but not obvious! It was proved by John Tate (1966) in his paper "Endomorphisms of Abelian Varieties over Finite Fields", p. 139. You can find the proof also in the ...
1 vote

1 vote

### Why weierstrass equation should not be used for elliptic curves over the field char=2?

Actually, the problem is the discriminant. By definition, an elliptic curve must have non-zero discriminant, otherwise it is degenerate. A clear explanation of this with examples of finite fields by ...
1 vote
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### Reductions of Isogenous Elliptic Curves

Let $f : E \to E'$ be an isogeny over $\mathbb{Q}$ and let $p$ be a prime of good reduction for $E, E'$. One way to see that the reductions (of global minimal integral Weierstrass models) \$\tilde{E}, \...

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