**Abstract**

We answer a question of Frey and Muller about whether or not the modular degree and congruence number of elliptic curves are equal. We give examples in which they are not, prove a theorem relating them, and make a conjecture about the extent to which they differ. We also obtain relations between analogues of the modular degree and congruence number for modular abelian varieties, and give new examples of failure of multiplicity one. |

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Update: The curve labeled 242B1 on page 3 of the paper should be replaced by 242a1 using "modern notation".