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We give an explicit construction of a class of F-theory models with matter inthe three-index symmetric (4) representation of SU(2). This matter is realizedat codimension two loci in the F-theory base where the divisor carrying thegauge group is singular; the associated Weierstrass model does not have theform associated with a generic SU(2) Tate model. For 6D theories, the matter islocalized at a triple point singularity of arithmetic genus g=3 in the curvesupporting the SU(2) group. This is the first explicit realization of matter inF-theory in a representation corresponding to a genus contribution greater thanone. The construction is realized by "unHiggsing" a model with a U(1) gaugefactor under which there is matter with charge q=3. The resulting SU(2) modelscan be further unHiggsed to realize non-Abelian G_2xSU(2) models with moreconventional matter content or SU(2)^3 models with trifundamental matter. TheU(1) models used as the basis for this construction do not seem to have aWeierstrass realization in the general form found by Morrison-Park, suggestingthat a generalization of that form may be needed to incorporate models witharbitrary matter representations and gauge groups localized on singulardivisors.

Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models