12
votes

### What is the taxicab number for rational fourth powers?

It has been known since Euler that the quartic surface defined by
$\displaystyle x_1^4 + x_2^4 = x_3^4 + x_4^4$
contains a rational curve defined by
$\displaystyle x_1(t) = t^7 + t^5 - 2t^3 + 3t^2 + t,...

9
votes

Accepted

### Family parametrizing elliptic curves with a rational 5-isogeny

Let $\phi:E\to E'$ be an isogeny of degree $5$, and $j, j'$ be the $j$-invariants of $E$ and $E'$, respectively. Let $r$ be the rational function $$r(z)=\frac{(z^2+10z+5)^3}{z}.$$ Then there is $\tau\...

2
votes

### When $E_D:y^2=x^3+17D^2x$ has even rank?

You can compute the root number over $\mathbb{Q}$ of any elliptic curve of the form $y^2=x^3+\alpha x$ for $\alpha\in \mathbb{Q}^*$ using the formulae in section 4B in
Density of rational points on ...

1
vote

### Elliptic curve points with the same $y$-coordinate

A bit too long for a comment: there is a (very) special case where this has a convenient answer.
Let $q$ be an odd prime power, and consider the elliptic curve $E/\mathbb{F}_q: y^3 = x^3 + b$ of $j$-...

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