ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 19 Jun 2020 05:03:12 -0500find characteristic polynomial in a tower of extensions over Qphttps://ask.sagemath.org/question/52087/find-characteristic-polynomial-in-a-tower-of-extensions-over-qp/We have a tower of extensions W/T/U where U is an unramified extension of $Q_p$, T is given by an eisenstein polynomial $f\in U[x]$ and W is given by an eisenstein polynomial $g\in T[x]$. Hence T is an $deg(g)deg(f)-$dimensional $U-$vector space with basis consisting of products $\alpha^i \beta^j$ such that $0\leq i < deg(f), 0\leq j < deg(g)$, where $\alpha$ is a root of $f$ and $\beta$ is a root of $g$.
Let $\gamma \in W$ be arbitrary. The goal is to compute the matrix of multiplication by $\gamma$ wrt the above $U-$basis. The issue is that the field $W$ cannot be created as an extension of $T$ in Sage, it seems that at the moment you cannot create an extension given by an eisenstein polynomial if T is already an eisenstein extension.
Example of how it does not work:
sage: U = Qq(2^2,names="u")<br>
sage: R.<x> = U[]<br>
sage: f = x^2 - U.uniformizer()<br>
sage: T = U.extension(f,names="alpha")<br>
sage: S.<x> = T[]<br>
sage: g = x^2-T.uniformizer()<br>
sage: W = T.extension(g,names="beta")<br>
TypeError: Unable to coerce -alpha to a rational<br>
We can create W as a quotient ring T[x]/(g), but I don't know how to get coefficients of $\gamma$ w.r.t. the power basis $1,\bar{x},\bar{x}^2,\dots,\bar{x}^{deg(g)-1}$.
sage: U = Qq(2^2,names="u")<br>
sage: R.<x> = U[]<br>
sage: f = x^2 - U.uniformizer()<br>
sage: T = U.extension(f,names="alpha")<br>
sage: S.<x> = T[]<br>
sage: g = x^2-T.uniformizer()<br>
**sage: W = S.quotient_ring(g)<br>
sage: V, map_to_W, map_from_W, = W.free_module()<br>**
NotImplementedError:<br>
If the above were possible, we could simply map $\gamma$ to the free module, then find coefficients wrt the power basis. Then we could do the same thing for each coefficient with W replaced by T to get the U-coefficients.
wltnFri, 19 Jun 2020 05:03:12 -0500https://ask.sagemath.org/question/52087/Real analytic Eisenstein serieshttps://ask.sagemath.org/question/51911/real-analytic-eisenstein-series/I'd to compute values of a certain function in Sage, a kind of a modular form, the so-called [real analytic Eisenstein series](https://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series). Does anybody know how to do it? I could not have found the Sage name for it.
More precisely, I would like to plot the the graph of the real analytic Eisenstein series, their real and imaginary values in a square of the complex plane (variable z) each for a certain value of the parameter s. Thus s is fixed in each of the graphs.
Real analytic Eisenstein series (also this is this their name at wikipedia) are defined for a complex z and a complex s. They are not the [Eisenstein series](https://en.wikipedia.org/wiki/Eisenstein_series) defined for complex z and an integer k. If s is k and an integer, they are connected by the multiple of Im(z)^s. Thus an easy connection, but I'd like to know the value of the series for a complex s. They are modular functions, not holomorphic and connected to theta series.svarotThu, 11 Jun 2020 13:40:09 -0500https://ask.sagemath.org/question/51911/