davidac897's profile - activity

Let's say I have an elliptic curve $E$ given by a Weierstrass equation in $x$ and $y$. Let's say I choose a uniformizer $t$ around a point $P$ on $E$.

Then is there a SAGE function that writes $x$ or $y$ as a power series in $t$ in a neighborhood of $P$?

Often, I might take something like $t=x$, so it's a matter of writing $y$.

Even more to the point, I want to take a differential form regular at $P$, say in the form $dx/y$ or $xdx/y$, and write it as a power series times $dt$?

More generally, for any smooth point on a curve, or even any smooth point on a scheme and a system of parameters at that point, there should be a function that takes a regular function in a neighborhood of that point and writes it as a power series in those parameters. I could imagine writing something like this myself, but it would take a lot of work, and I would hope this is already implemented.

In the SAGE Reference Manual, there's a brief section on graded Hopf algebras: https://doc.sagemath.org/html/en/refe...

Can one actually define graded Hopf algebras and do computations in them? If not, what is this doing there?

I have a cubic number field $M$, and I want to find the norm form of its integer ring (as a degree $3$ polynomial in $3$ variables). Does this functionality exist in SAGE?

For now, I found this solution:

O = M.maximal_order()
RRR.<a,b,c> = PolynomialRing(M)
lists = []
d = 1

for i in O.gens():
    lists.append(i.galois_conjugates(M))

for i in range(3):
    e = 0
    e += a*lists[0][i]
    e += b*lists[1][i]
    e += c*lists[2][i]
    d *= e

d

Let's say I have a vector space $V$ of dimension $n$, and I have various elements of $V \otimes V$, which you can think of as $n \times n$ matrices. I want to check whether these elements are linearly dependent, and in some cases find a relation. But if I create a matrix space MS, it has no attribute linear_dependence, like with vector spaces.

Do I have to just create a $n^2$-dimensional vector space $W$ and then define the bilinear mapping from $V$ to $W$? And what about rank $3$ tensors, which form a $n^3$-dimensional space?

Suppose $K$ is a number field, $\mathfrak{m}$ is a modulus of $K$, and $\mathfrak{a}$ is a given fractional ideal of $K$. SAGE can compute the ray class group $Cl_{\mathfrak{m}}(K)$. However, how can I find what element of the ray class group $\mathfrak{a}$ corresponds to? More specifically, I need to find the order of $\mathfrak{a}$ in $Cl_{\mathfrak{m}}$.

I can do this in MAGMA only when $\mathfrak{m}=(1)$, but does anyone know of a way to do this in SAGE?