ellgenerators

Function: ellgenerators
Section: elliptic_curves
C-Name: ellgenerators
Prototype: G
Help: ellgenerators(E): if E is an elliptic curve over the rationals,
 return the generators of the Mordell-Weil group attached to the curve.
 This relies on the curve being referenced in the elldata database.
 If E is an elliptic curve over a finite field Fq as output by ellinit(),
 return a minimal set of generators for the group E(Fq).
Doc:
 If $E$ is an elliptic curve over the rationals, return a $\Z$-basis of the
 free part of the \idx{Mordell-Weil group} attached to $E$.  This relies on
 the \tet{elldata} database being installed and referencing the curve, and so
 is only available for curves over $\Z$ of small conductors.
 If $E$ is an elliptic curve over a finite field $\F_q$ as output by
 \tet{ellinit}, return a minimal set of generators for the group $E(\F_q)$.

 \misctitle{Caution} When the group is not cyclic, of shape $\Z/d_1\Z \times
 \Z/d_2\Z$ with $d_2\mid d_1$, the points $[P,Q]$ returned by ellgenerators
 need not have order $d_1$ and $d_2$: it is true that
 $P$ has order $d_1$, but we only know that $Q$ is a generator of
 $E(\F_q)/<P>$ and that the Weil pairing $w(P,Q)$ has order $d_2$,
 see \kbd{??ellgroup}.
 If you need generators $[P,R]$ with $R$ of order $d_2$, find
 $x$ such that $R = Q-[x]P$ has order $d_2$ by solving
 the discrete logarithm problem $[d_2]Q = [x]([d_2]P)$ in a cyclic group of
 order $d_1/d_2$. This will be very expensive if $d_1/d_2$ has a large
 prime factor.