but more explicitly one obtains the Hodge numbers
h2,0=1,h1,1=4,h0,2=1,h^{2,0}=1,\qquad h^{1,1}=4,\qquad h^{0,2}=1,h2,0=1,h1,1=4,h0,2=1,
and so dimCH1,1(X)=4\dim_{\mathbb C}H^{1,1}(X)=4dimCH1,1(X)=4 (note dimQ(H1,1H2(X,Q))\dim_{\mathbb Q} (H^{1,1}\cap H^2(X,\mathbb Q))dimQ(H1,1H2(X,Q)) may be smaller, depending on rational structures).
2. Algebraic cycles: divisors and their classes
On the surface X=EEX=E\times EX=EE there are immediate algebraic divisors of elementary geometric origin:
Horizontal and vertical fibres:
Dh:=E{p},Dv:={p}E,D_h \;:=\; E\times\{p\},\qquad D_v \;:=\; \{p\}\times E,Dh:=E{p},Dv:={p}E,
for any point pEp\in EpE. The cohomology classes [Dh][D_h][Dh] and [Dv][D_v][Dv] are (integral) elements of H1,1(X)H2(X,Z)H^{1,1}(X)\cap H^2(X,\mathbb{Z})H1,1(X)H2(X,Z).
The diagonal EE\Delta\subset E\times EEE:
={(x,x)EE},\Delta \;=\; \{(x,x)\in E\times E\},={(x,x)EE},
whose cohomology class [][\Delta][] is algebraic and lies in H1,1(X)H^{1,1}(X)H1,1(X). (Equivalently one may consider DhDv\Delta - D_h - D_vDhDv to obtain primitive classes.)
These divisors generate the Nron--Severi group NS(X)=Pic(X)/Pic0(X)\operatorname{NS}(X) = \operatorname{Pic}(X)/\operatorname{Pic}^0(X)NS(X)=Pic(X)/Pic0(X) as a lattice (over Z\mathbb ZZ) for the generic product of elliptic curves; in particular, the divisor classes span a rational subspace of H1,1(X)H^{1,1}(X)H1,1(X).
By the Lefschetz (1,1)(1,1)(1,1)-theorem (valid for compact Khler manifolds and hence for smooth projective varieties), every integral class in H1,1(X)H2(X,Z)H^{1,1}(X)\cap H^2(X,\mathbb Z)H1,1(X)H2(X,Z) is the class of a divisor. Consequently,
H1,1(X)H2(X,Q)=Im(cl1:CH1(X)QH2(X,Q)).H^{1,1}(X)\cap H^2(X,\mathbb{Q}) \;=\; \operatorname{Im}\big( \operatorname{cl}_1 : \mathrm{CH}^1(X)\otimes\mathbb{Q}\to H^2(X,\mathbb{Q})\big).H1,1(X)H2(X,Q)=Im(cl1:CH1(X)QH2(X,Q)).
Thus for codimension p=1p=1p=1 the Hodge Conjecture holds: every rational Hodge class of type (1,1)(1,1)(1,1) is algebraic.
3. Explicit basis and the cycle class map
Choose points p,qEp,q\in Ep,qE. Then the divisor classes
