Determining $\mu$-invariant of elliptic curves over $\mathbb{Q}$

Question

From Pollack's table on his homepage, I have the values of mu invariant of elliptic curves 38B1 & 38B2 (labeled as in Cremona table). But I need to know the values of mu invariants of 38A1, 38A2, 38A3. Also please inform me, how to calculate the mu invariants of elliptic curves in general.

Answer 1

At what prime ? Well, let $p$ be an odd prime at which $E$ has good ordinary reduction. There are two ways:

• Use sage or magma to compute the analytic $p$-adic L-function via modular symbols. In sage this is E.padic_lseries(p).series(n) where $n$ is the precision to which the series is computed. The $\mu$-invariant is often easy to read off the first few terms with a small $n$. That is what Robert Pollack did.

• Assume the conjecture by Greenberg that the $\mu$-invariant is zero for the minimal curve. Then you know immediately that the $\mu$-invariant is zero for all such primes $p$ except the ones for which $E$ has an isogeny of that degree. Once you know the $\mu$-invariant for the minimal curve it is easy to compute it for all others as it just changes with the quotient of the periods.

For the isogeny class 38a, the $\mu$-invariant should be trivial for all good ordinary primes $p>3$. For $p=3$, it is $\mu=0$ for 38a3, $\mu=1$ for 38a1 and $\mu=2$ for 38a2.