Returns a list with the following items:
OmegaA complex vector of length 2 giving the fundamental half
periods \(\omega_1\) and \(\omega_2\). Notation
follows Chandrasekharan: half period
\(\omega_1\) is 0.5 times a (nontrivial) period of minimal
modulus, and \(\omega_2\) is 0.5 times a period of smallest
modulus having the property \(\omega_2/\omega_1\)
not real.
The relevant periods are made unique by the further requirement that
\(\mathrm{Re}(\omega_1)>0\), and
\(\mathrm{Im}(\omega_2)>0\); but note that this
often results in sign changes when considering cases on boundaries
(such as real \(g_2\) and \(g_3\)).
Note Different definitions exist for \(\omega_3\)!
A and S use \(\omega_3=\omega_2-\omega_1\),
while Whittaker and Watson (eg, page 443), and Mathematica, have
\(\omega_1+\omega_2+\omega_3=0\)
qThe nome. Here,
\(q=e^{\pi i\omega_2/\omega_1}\).
gComplex vector of length 2 holding the invariants
eComplex vector of length 3. Here \(e_1\), \(e_2\),
and \(e_3\) are defined by
$$e_1=\wp(\omega1/2)m\qquad e_2=\wp(\omega2/2),\qquad
e_3=\wp(\omega3/2)$$
where \(\omega_3\) is defined by
\(\omega_1+\omega_2+\omega_3=0\).
Note that the \(e\)s are also defined as the three roots of
\(x^3-g_2x-g_3=0\); but this method cannot be used in
isolation because the roots may be returned in the wrong order.
DeltaThe quantity \(g_2^3-27g_3^2\), often
denoted \(\Delta\)
EtaComplex vector of length 3 often denoted
\(\eta\). Here
\(\eta=(\eta_1,\eta_2,\eta_3)\) are defined
in terms of the Weierstrass zeta function with
\(\eta_i=\zeta(\omega_i)\) for \(i=1,2,3\).
Note that the name of this element is capitalized to avoid confusion
with function eta()
is.AnSBoolean, with TRUE
corresponding to real
invariants, as per Abramowitz and Stegun
givencharacter string indicating which parameter was supplied.
Currently, one of “o
” (omega), or “g
”
(invariants)