With 2^2 isogeny it never works:
- F. Richelot, De transformatione integralium Abelianorum primi ordinis comentatio. J. reine angew. Math. 16 (1837) 221-341
- G. Humbert, Sur la transformation ordinaire des fonctions abeliennes. J. de Math. (5) 7 (1901), 359 - 417 )
- J.-B. Bost and J.-F. Mestre, Moyenne arithmetico-geometrique et periodes des courbes de genre 1 et 2, Gaz. Math. 38 (1988), 36-64
With 3^2 it always does (unless the kernel of the isogeny splits on the two components) - see 3.3 in http://arxiv.org/abs/0710.1298 ( = AMS contemp. math. series, volume 465 (2008): Curves and abelian varieties, p 51-69)