TITLE: What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?
QUESTION [12 upvotes]: There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered if anyone can shed any more light on the problem for me.
Let $p\geq 5$ be a prime and let $C=C_p: y^2  = x^3 + 1$ be the cubic curve over the finite field $\mathbb{F}_p$. Let $M_p = \left\vert C(\mathbb{F}_p)\right\vert$ be the order of the group of rational points of $C$ (this includes the point at infinity). Looking at some small primes I proved that if $p\equiv 2 \pmod 3$ then $M_p = p+1$.
The interesting case is when $p\equiv 1 \pmod 3$. I wrote a program which computes $M_p$ where $p\equiv 1\pmod 3$ is prime. We get the following sequence, where the $i$th term is the value of $M_p$ for the $i$th largest prime $p$ of the form $p\equiv 1\pmod 3$, and I have computed here for all $p\leq 1000$:
$12, 12, 12, 36, 48, 36, 48, 84, 84, 84, 84, 84, 108, 108, 156, 156, 144, 156, 156, 192, 228, 228, 252, 252, 228, 300, 252, 252, 324, 336, 300, 372, 336, 372, 336, 372, 432, 372, 444, 432, 468, 468, 444, 444, 468, 516, 588, 588, 588, 624, 576, 588, 624, 588, 588, 684, 624, 624, 684, 732, 684, 684, 756, 804, 732, 768, 732, 756, 876, 876, 912, 804, 912, 876, 948, 972, 912, 948, 948, 1008, \dots$
It seems that however many terms you take their GCD is always $12$, and this doesn't change as far as $p\leq 3000$, so I have two questions:

Can anyone prove that $M_p$ is always divisible by 12?
Since 1992 when the book was published, has anyone found an expression for $M_p$?

Any extra information about this sequence (or related topics/papers) is very welcome!

REPLY [4 votes]: If $p$ is $\equiv 1 \bmod 3,$ then we may write $p = a^2 + 3 b^2$, where $a$ is pinned down by requiring $a \equiv 1 \bmod 3$ (and I won't bother to pin down $b$).
Then, writing $a_p = 1 + p -  | C(\mathbb F_p)|$, as usual, one has that
$$a_p = 2a,$$
or equivalently,
$$| C(\mathbb F_p)| = 1 + p - 2 a .$$
In terms of Qiaochu's answer, the relevant modular form is a CM modular forms, and so can be expressed in terms of a Hecke character for the CM field (which
is $\mathbb Q(\zeta_3)$ in this case).
More directly, the curve $C$ is a CM elliptic curve, with CM by $\mathbb Z[\zeta_3]$ (the element $\zeta_3$ acts by $(x,y) \mapsto (\zeta_3 x, y)$), and so it corresponds to a Hecke character of $\mathbb Q(\zeta_3)$.  Explicitly computing this Hecke character gives the above result.