Dear MO,

Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us choose $K/\mathbb{Q}_p$ of minimal degree such that $E/K$ has good reduction, and let us assume $E/K$ is given by a minimal model. Let $A$ be the ring of integers of $K$, let $\pi$ be a uniformizer for $A$, let $\nu$ be a valuation on $K$ with $\nu(\pi)=1$ and $\nu(p)=e$, and let $\hat{E}/A$ be the formal group associated to $E/K$.

Let $\[p](X) = pX+\cdots=\sum_{k=1}^\infty a_kX^k$ be the formal power series for the multiplication-by-$p$ map on $\hat{E}$. Let $e_0=\nu(a_1)=\nu(p)=e$, and $e_1=\nu(a_p)$. Since $E/K$ is supersingular, we know that $e_2=\nu(a_{p^2})=0$.

If we define points in the plane by $P_0=(1,e)$, $P_1=(p,e_1)$ and $P_2=(p^2,0)$, then $N$, the Newton polygon of $[p](X)$ (for those roots with valuation $>0$) is given either by one single segment $P_0P_2$, or two segments $P_0P_1$ and $P_1P_2$, according to whether $ep/(p+1) \leq e_1$ or $ep/(p+1) > e_1$, respectively.

The number $e$ is a divisor of $12$, and if $p\geq 5$, then $e\leq 6$. If $e>1$, and $N$ has two segments, then $1\leq e_1 < e$. Suppose that we are in this case, i.e., $e_1 < e$.

**Question**: Are there any further constraints on the values of $e_1$?

More specifically:

**Question 1**: Are there any divisibility conditions on $e_1$, or further relations between $e$ and $e_1$? Can $e_1$ take any of the values $1\leq e_1< e$ ?

**Question 2**: Is there a more conceptual way to think about $e_1$?

In all examples I have computed, I get that $e_1=1$, $2$ or $4$ (but I haven't done an exhaustive search either). For instance:

$E=27a4,\quad p=3, \quad [K:\mathbb{Q}]=12, \quad e=12, \quad e_1=2;$

$E=121a2,\quad p=11, \quad K=\mathbb{Q}(\sqrt[6]{11}), \quad e=6, \quad e_1 = 4;$

$E=14450p2,\quad p=17, \quad K=\mathbb{Q}(\sqrt[3]{17}), \quad e=3, \quad e_1=1.$

Thank you in advance for any answers!

Alvaro