Weight $144169$

Barry Mazur's 144169 problem (see the two-page pdf that Mazur sent me).

Remark 4.3.1 (From Barry Mazur.)  

I've just glanced (fast) through the problem book and I wanted to send you a note telling you that I think it is wonderful. It really seems enticing, rich, friendly, and vastly interesting. About the 144169 problem, there was a bit of discussion about it and related things last Spring (Kevin Buzzard, and Robert Pollack, in particular, had ideas). The more general question behind the 144169 example is to consider $\mathbb{T}= \mathbb{T}_p$ the p-adic (Hida) Hecke algebra (say, of tame level 1, for starters) as a $\Lambda$ -algebra and to form D = discriminant of the finite flat $\Lambda$ -algebra $\mathbb{T}$ , so that $D$ is in $\Lambda$ (i.e., is an Iwasawa function). We want to know something about the basic invariants of $D$ , e.g., its "$\lambda$ -invariant " in each of the $p-1$ discs that form the rigid-analytic space underlying $\Lambda= \mathbb{Z}_p[[\mathbb{Z}_p^*]]$ and more specifically, we want to know something about the placement of the zeroes of $D$ , if there are any. With p=144169 and the 24th disc, since that part of $\mathbb{T}$ is quadratic over that part of $\Lambda$ , there could be some zeroes, so the question is: are there some, and how many? If I remember right, Kevin had an idea about how to quickly compute this and Pollack had an idea of how--in the context of some tame level--to get examples for low primes like $p=3$ and $p = 5$ , where $D$ has some zeroes.