h2,2(E4)=6.h^{2,2}(E^4) \;=\; 6.h2,2(E4)=6.
Equivalently, one may enumerate the pure-type tensors
ijk,{i,j,k,}={1,2,3,4},\alpha_{i}\wedge\alpha_{j}\otimes \overline\alpha_{k}\wedge\overline\alpha_{\ell}, \qquad \{i,j,k,\ell\}=\{1,2,3,4\},ijk,{i,j,k,}={1,2,3,4},
where m\alpha_mm denotes the holomorphic 111-form on the mmm-th factor and m\overline\alpha_mm its conjugate. Each unordered choice {i,j}\{i,j\}{i,j} of two factors determines a unique (up to scalar) basis element of H2,2(X)H^{2,2}(X)H2,2(X).
3. Algebraic generators: products of point classes (product of divisors)
An elliptic curve EEE has algebraic divisors of codimension 111 given by points. On X=E4X=E^4X=E4, a codimension 222 algebraic cycle may be produced by taking the product of point-classes on two chosen factors and taking the whole fiber EEE on the remaining factors. Concretely, for indices 1i<j41\le i<j\le 41i<j4 and fixed points piEi,pjEjp_i\in E_i, p_j\in E_jpiEi,pjEj, the subvariety
Zi,j:=E1Ei1{pi}Ei+1Ej1{pj}Ej+1E4Z_{i,j} \;:=\; E_1\times\cdots\times E_{i-1}\times\{p_i\}\times E_{i+1}\times\cdots\times E_{j-1}\times\{p_j\}\times E_{j+1}\times\cdots\times E_4Zi,j:=E1Ei1{pi}Ei+1Ej1{pj}Ej+1E4
is a codimension-2 algebraic cycle in XXX. The class [Zi,j][Z_{i,j}][Zi,j] lies in CH2(X)\mathrm{CH}^2(X)CH2(X) and its image under the cycle class map is a class in H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb Q)H2,2(X)H4(X,Q).
There are exactly (42)=6\binom{4}{2}=6(24)=6 independent such product cycles (up to rational equivalence and for suitably generic choices of points), one for each unordered choice {i,j}\{i,j\}{i,j}. These product cycles furnish six algebraic classes which we denote z12,z13,z14,z23,z24,z34z_{12}, z_{13}, z_{14}, z_{23}, z_{24}, z_{34}z12,z13,z14,z23,z24,z34.
4. Identification of Knneth basis with product cycles
Under the Knneth isomorphism, the abstract basis element determined by the choice {i,j}\{i,j\}{i,j} (two (1,0)(1,0)(1,0)-factors on i,ji,ji,j and (0,1)(0,1)(0,1)-factors on the complement) corresponds, up to a nonzero scalar, to the class [Zi,j][Z_{i,j}][Zi,j] obtained by taking points on factors iii and jjj. Thus there is a natural bijection between the Knneth indexing of H2,2(X)H^{2,2}(X)H2,2(X) and the list of product-of-point cycles {[Zi,j]}\{[Z_{i,j}]\}{[Zi,j]}. Consequently, the six classes [Zi,j][Z_{i,j}][Zi,j] form a Q\mathbb QQ-basis of H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb Q)H2,2(X)H4(X,Q).
