elleisnum

Function: elleisnum
Section: elliptic_curves
C-Name: elleisnum
Prototype: GLD0,L,p
Help: elleisnum(w,k,{flag=0}): k being an even positive integer, computes the
 numerical value of the Eisenstein series of weight k at the lattice
 w, as given by ellperiods. When flag is nonzero and k=4 or 6, this gives the
 elliptic invariants g2 or g3 with the correct normalization.
Doc: $k$ being an even positive integer, computes the numerical value of the
 Eisenstein series of weight $k$ at the lattice $w$, as given by
 \tet{ellperiods}, namely
 $$
 (2i \pi/\omega_2)^k
 \Big(1 + 2/\zeta(1-k) \sum_{n\geq 1} n^{k-1}q^n / (1-q^n)\Big),
 $$
 where $q = \exp(2i\pi \tau)$ and $\tau:=\omega_1/\omega_2$ belongs to the
 complex upper half-plane. It is also possible to directly input $w =
 [\omega_1,\omega_2]$, or an elliptic curve $E$ as given by \kbd{ellinit}.
 \bprog
 ? w = ellperiods([1,I]);
 ? elleisnum(w, 4)
 %2 = 2268.8726415508062275167367584190557607
 ? elleisnum(w, 6)
 %3 = -3.977978632282564763 E-33
 ? E = ellinit([1, 0]);
 ? elleisnum(E, 4)
 %5 = -48.000000000000000000000000000000000000
 @eprog

 When \fl\ is nonzero and $k=4$ or 6, returns the elliptic invariants $g_2$
 or $g_3$, such that
 $$y^2 = 4x^3 - g_2 x - g_3$$
 is a Weierstrass equation for $E$.
 \bprog
 ? g2 = elleisnum(E, 4, 1)
 %6 = -4.0000000000000000000000000000000000000
 ? g3 = elleisnum(E, 6, 1)  \\ ~ 0
 %7 = 0.E-114 - 3.909948178422242682 E-57*I
 @eprog