  The Gamow-Teller and Fermi matrix elements, ${\left<\mathbf{GT}\right>}$ and
${\left<\mathbf{F}\right>}$, respectively, for tritium $\beta$-decay are
calculated to next-to-leading order (NLO) in pionless effective field theory in
the absence of Coulomb and isospin violation giving the leading order
predictions ${\left<\mathbf{GT}\right>}_{0}=0.9807$ and
${\left<\mathbf{F}\right>}_{0}=1$. Using an experimentally determined value for
the tritium-$\beta$ decay GT matrix element, the two-body axial current low
energy constant is fixed at NLO yielding $L_{1,A}=6.01\pm2.08$ fm$^{3}$ at the
renormalization scale of the physical pion mass, which agrees with predictions
based on naive dimensional analysis. Finally, the consequences of Wigner-SU(4)
spin-isospin symmetry are considered for the Gamow-Teller matrix element.
