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Let$UY_{n}(q)$be a Sylow$p$-subgroup of an untwisted Chevalley group$Y_{n}(q)$of rank$n$defined over $\mathbb{F}_{q}$where$q$is a power of a prime$p$. We partition the set$\text{Irr}(UY_{n}(q))$of irreducible characters of$UY_{n}(q)$into families indexed by antichains of positive roots of the root system of type$Y_{n}$. We focus our attention on the families of characters of$UY_{n}(q)$which are indexed by antichains of length$1$. Then for each positive root$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.

In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.