TITLE: BSD conjecture for X_0(17)
QUESTION [7 upvotes]: I use Magma to calculate the L-value, yields
E:=EllipticCurve([1, -1, 1, -1, 0]);
E;
Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E);
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - x over Rational Field
0.386769938387780043302394751243 3.09415950710224034641915800995
0.125000000000000000000000000000
$#torsionsubgroup = 4, c_{17}(E)=1.$
But the strong BSD predicts that
$L(E,1)/\Omega_{\infty}$= $(#Sha(E)/#tor(E)^2)*c_{17}(E)$
We will get $L(E,1)/\Omega_{\infty}=1/16$, not $1/8$.
Why does that happen? Thanks a lot.

REPLY [16 votes]: Tim's answer is great, but I want to mention one other place where people have lost a power of 2. The canonical height is often defined relative to the divisor (O), as I do in my books. So it is given by
$$ \hat h(P) = \frac12 \lim_{n\to\infty}  4^{-n} h\bigl(x([2^n]P)\bigr). $$
Here the $\frac12$ is inserted because $x$ has a double pole at $\infty$. However, in computing the height regulator for BSD, one should not use the $\frac12$, i.e., one should compute heights relative to the divisor $2(O)$. 
Why, one might ask, use a weird divisor like $2(O)$. The answer is that when BSD is properly formulated for abelian varieties, it uses the height pairing $A(K)\times \hat A(K)$ that pairs points of $A$ with points on its dual, and the pairing is done relative to the Poincare divisor on $A\times \hat A$. If one traces through the definitions and identifies an elliptic curve with its dual, the Ponicare divisor on $E\times E$ is $(O)\times E + E\times (O)$, which eventually shows that one should use the height on $E$ relative to $2(O)$.
Note that this means that if you compute BSD using the wrong height on a curve of rank $r$, then your answer will be off by a factor of $2^r$.
(I learned about this potential error from Dick Gross many years ago.)