Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e.
$$ [j, HZ/p]\_\*$$
as a module over Steenrod algebra. Or dually the homology of $j$ as a comodule over the dual Steenrod algebra. Is there a nice description? 


A reference will be appreciated as well. 



Cohomology of the image of J spectrum
This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is
(at p=2) $H^\* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota\_0$, $Sq^2 \iota\_0$, $Sq^4\iota\_0$, $Sq^8\iota\_0 + Sq^1 \iota\_7$, $Sq^7\iota\_7$, and $(Sq^{(0,1,1)} + Sq^{(4,2)})\iota\_7$, in Milnor basis notation. If you prefer the admissable basis, this last operation on $\iota\_7$ is $
Sq^6 Sq^3 Sq^1 + Sq^7 Sq^2 Sq^1 + Sq^7 Sq^3 + Sq^8 Sq^2 + Sq^9 Sq^1 + Sq^{10}$.


This was also computed in MR2171809 Angeltveit, Vigleik ; Rognes, John . Hopf algebra structure on topological Hochschild homology. Algebr. Geom. Topol. 5 (2005), 1223--1290.


I derive this quite explicitly and elementarily, as a sample of how to use some sage code Mike Catanzaro wrote, in a talk I gave at Northwestern, "http://www.rrb.wayne.edu/papers/jnwuhand.pdf", where I also give more precise citation of Don Davis's paper.


Briefly,


1. you observe that $\psi^3 - 1$ must induce $Sq^4 : H^\* \Sigma^4 ksp \to H^\* ko$,
2. you compute the ker and coker
3. compute that $Ext^1 = F\_2$, and that the split extension would give the wrong homotopy,
4. compute the non-trivial extension using a cocycle defining it.


There, I also note that the Adams spectral sequence for this is quite clean: a marvelous $d\_2$ clears out all the rubbish, followed by a simple sequence of differentials which follow from the Leibniz rule, to give the groups of order the 2-adic valuation of 8j in dimensions 8j-1.


Added:
In that talk, I used a conjecture that a certain homomorphism really does induce $d\_2$. That has been proved by John Rognes. See <https://arxiv.org/abs/2105.02601> (has now appeared in Trans AMS).


Also, there is an interesting module over A(2) that comes up in the j/2 case. The module exhibits periodicity, $\Omega^4 M \simeq \Sigma^{12}M$, and has Ext exactly equal to the associated graded of the homotopy of j/2, in filtrations 2 and up. It is defined over $A(2)$.


