696.e1 Base Certificate

Writes the complete base data of 696.e1 into a formal certificate: $E:y^2=x^3+x^2+8x-16$, $N=696=2^3\cdot3\cdot29$, $\Delta_{\min}=-2^{11}\cdot3\cdot29<0$, $E(\mathbb Q)_{\rm tors}=0$, both analytic/algebraic rank are $0$, optimal, Manin constant $1$. The document then does something very important: because the introduction of Banwait–Huang recalls that full BSD has been rigorously verified for analytic-rank-0/1 elliptic curves with conductor $\le5000$, and $696<5000$, $r_{\rm an}=0$, therefore $\operatorname{BSD}(E)$ — especially $\operatorname{BSD}(E,2)$ — is a rigorous result directly inherited from existing literature, not newly proven in this series; the document explicitly notes that it deliberately avoids the circular reasoning of "analytic Sha=1 implies actual Sha=1" here. For the $L^{alg}$ gate part, LMFDB records $\Sha_{\rm an}=1$, $\prod c_p=1$, $|E(\mathbb Q)_{\rm tors}|=1$, $Reg=1$, and from the definition of analytic Sha we directly obtain $v_2(L^{alg}(E,1))=0$. Finally, it calculates the 2-division cubic: since $a_1=a_3=0$, $f_2(x)=x^3+x^2+8x-16$, which has no rational roots and is thus irreducible. Its discriminant $\operatorname{disc}(f_2)=-11136=-2^7\cdot3\cdot29$ is not a square, so the Galois closure is $S_3$, and the quadratic resolvent is $\mathbb Q(\sqrt{-174})$ — this set of calculations directly paves the way for the Chebotarev argument in document 16.