In this talk we will focus on constructing "super-isolated abelian varieties". These are abelian varieties that have isogeny class which contains a single isomorphism class. Their motivation comes from security concerns in elliptic and hyperelliptic curve cryptography. Using a theorem of Honda and Tate, we transfer the problem of finding such varieties to a problem in algebraic number theory. Finding these varieties turns out to be related to finding primes of the form n2 + 1 and to solving Pell's equation.