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.



BSD conjecture for X_0(17)
To expand my comment, there are at least 3 subtle ways to get BSD wrong:


1) The BSD period over ${\mathbb Q}$ is the real period $\Omega\_\infty$ when $E({\mathbb R})$ is connected ($\Delta(E)<0$) and $2\Omega\_\infty$ when it has two connected components ($\Delta(E)>0$). The same thing happens over number fields, at every real place. So in your example you have to divide the $L$-value by $2\Omega\_\infty$ to get $1/16$.


Another way (alternative) of phrasing this: is that the Tamagawa number at an Archimedean place is $2$ if real and split ($\Delta>0$), and $1$ otherwise. It is less easier to forget to include it then, and you can always use $\Omega\_\infty$. Magma does not have Tamagawa numbers at infinite places directly.


The other two concern BSD over number fields:


2) The height pairing in BSD depends on the ground field. For instance, $E=37A1$ has $E({\mathbb Q})={\mathbb Z}\cdot P$ and $E({\mathbb Q(i)})={\mathbb Z}\cdot P$, where $P$ is the same point $(0,0)$. The regulator is $\approx 0.051$ over ${\mathbb Q}$ but $\approx 2\cdot 0.051=0.102$ over ${\mathbb Q(i)}$; the factor $2$ is $[{\mathbb Q(i)}:{\mathbb Q}]$.


3) Over ${\mathbb Q}$ there is this luxury of having a global everywhere minimal model, so there is a canonical differential to integrate. Over number fields you cannot do this, so one usually takes any invariant differential and introduces a correction term that measures its failure to be minimal at all primes. The point is that if you start with a curve over ${\mathbb Q}$ with additive reduction at $p$ and go up to a number field $K$ where $p$ ramifies, e.g. $50A3$ over $K={\mathbb Q}(\sqrt 5)$, the minimal model might stop being minimal, and this correction factor comes in for BSD over $K$. 


The functions ConjecturalRegulator and ConjecturalSha in Magma take care of these normalizations - it's actually quite nice to experiment with them. 


Hope this helps. 


P.S. You would not believe how many times each of these mistakes was made!


