Omori-Utsu law. Aftershock rate n(t) = K / (t + c)^p (events per day). The expected count between t1 and t2 is K [(t1+c)^(1-p) - (t2+c)^(1-p)] / (p-1); here K is set from the productivity N365 = 0.5 x 10^(Mw - Mc - 1) events above the completeness magnitude Mc in the first year (illustrative productivity, real sequences vary by a factor of ten).
Gutenberg-Richter. log10 N(>=M) = a - b M. Magnitudes are drawn from the truncated exponential law above Mc, capped just below the mainshock. Aki's maximum-likelihood estimate is b = log10(e) / (mean(M) - (Mc - dM/2)) with binning dM, and has standard error b / sqrt(n).
Bath's law. The expected largest aftershock sits where the expected count above it is one: M1 = Mc + log10(N)/b plus 0.25/b (mean of the extreme-value law). For the default scenario this is about 1.3 below the mainshock.
Energy. log10 E = 1.5 M + 4.8 (E in joules): one magnitude step is a factor 31.6 in energy, so one M7 releases as much as about 1000 M5 events.
Rupture length. log10 L = -2.44 + 0.59 Mw (km), a typical regression for surface rupture length; aftershocks scatter around the fault trace. Shaking. The intensity rings use the illustrative law I = 1.5 + 1.5 Mw - 3.5 log10 R - 0.003 R (R in km) with geometric spreading and anelastic loss; real ground motion depends strongly on site and depth. Constants are teaching values quoted from memory, to be spot-checked; this is not a hazard tool.