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Galaxy lenses are frequently modeled as an elliptical mass distribution withexternal shear and isothermal spheres to account for secondary andline-of-sight galaxies. There is statistical evidence that some fraction ofobserved quads are inconsistent with these assumptions, and require adipole-like contribution to the mass with respect to the light. Simplifyingassumptions about the shape of mass distributions can lead to the incorrectrecovery of parameters such as $H_0$. We create several tests of synthetic quadpopulations with different deviations from an elliptical shape, then fit themwith an ellipse+shear model, and measure the recovered values of $H_0$.Kinematic constraints are not included. We perform two types of fittings -- onewith a single point source and one with an array of sources emulating anextended source. We carry out two model-free comparisons between our mock quadsand the observed population. One result of these comparisons is a statisticalinconsistency not yet mentioned in the literature: the image distance ratioswith respect to the lens center of observed quads appear to span a much widerrange than those of synthetic or simulated quads. Bearing this discrepancy inmind, our mock populations can result in biases on $H_0$ $\sim10\%$.

Galaxy-lens determination of $H_0$: the effect of the ellipse+shear modeling assumption