Function: ellbsd
Section: elliptic_curves
C-Name: ellbsd
Prototype: Gp
Help: ellbsd(E): E being an elliptic curve over a number field,
 returns a real number c such that the BSD conjecture predicts that
 lfun(E,1,r)/r! = c*R*S where r is the rank, R is the regulator and S is the
 cardinal of the Tate-Shafarevich group.
Doc:
 $E$ being an elliptic curve over a number field, returns a real
 number $c$ such that the Birch and Swinnerton-Dyer conjecture predicts that
 $L_{E}^{(r)}(1)/r!{} = c\*R\*S$, where $r$ is the rank, $R$ the regulator and
 $S$ the cardinal of the Tate-Shafarevich group.

 \bprog
 ? e = ellinit([0,-1,1,-10,-20]); \\ rank 0
 ? ellbsd(e)
 %2 = 0.25384186085591068433775892335090946105
 ? lfun(e,1)
 %3 = 0.25384186085591068433775892335090946104
 ? e = ellinit([0,0,1,-1,0]); \\ rank 1
 ? P = ellheegner(e);
 ? ellbsd(e)*ellheight(e,P)
 %6 = 0.30599977383405230182048368332167647445
 ? lfun(e,1,1)
 %7 = 0.30599977383405230182048368332167647445
 ? e = ellinit([1+a,0,1,0,0],nfinit(a^2+1)); \\ rank 0
 ? ellbsd(e)
 %9 = 0.42521832235345764503001271536611593310
 ? lfun(e,1)
 %10 = 0.42521832235345764503001271536611593309
 @eprog
