  We consider anomaly cancellation for $SU(N)\times SU(2)\times U(1)$ gauge
theories where the left-handed chiral multiplets are in higher $SU(2)$
representations. In particular, if the left-handed quarks and leptons transform
under the triplet representation of $SU(2)$ and if the $U(1)$ gauge group is
compact then up to an overall scaling there is only one possible nontrivial
assignment for the hypercharges if $N=3$, and two if $N=9$. Otherwise there are
infinitely many. We use the Mordell-Weil theorem, Mazur's theorem and the
Cremona elliptic curve database which uses Kolyvagin's theorem on the Birch
Swinnerton-Dyer conjecture to prove these statements.
