I'd like to expand on the oscillator bias schema shown.
The collector of Q1 is biased using a choke inductor which has 0V drop at DC but presents ~370Ω impedance at 27MHz (Z_L=ωxL). The base is biased with 100kΩ (though it's in parallel with βxR_emitter, which is still in 10s of kΩs). The emitter is degenerated by 100Ω. Based on basic BJT equations (already in other answers) and C945s datasheet (for values of beta), we can estimate that the Q1 Vb and Vc are at about 4.5V, 3.8V respectively, and collector/emitter current is about 5-8mA.
Also because the Q1 collector sees a much lower impedance at 27MHz than the base, I believe that the main oscillation signal is generated at the base, then transferred to the emitter. Then a fraction of the emitter signal (about 1/6 or so, based on current division between R8 & C4+L3) is coupled by C4 to the collector and then adds back to the base signal through the crystal which is very high impedance at resonant frequency of 27MHz (due to the inherent property of the resonance tanks).
I believe that this oscillator is a base-to-emitter current amplifier. At risk of responding to comments in the answer, the idea proposed in one of the previous comments that this is a common-base amplifier with input at collector and output at emitter is likely not feasible. In that configuration, the current would be barely amplified (1.001 times?), and voltage signal would be significanly attenuated, thus not meeting the criteria for oscillations given the losses in aforementioned 1/6 current division in the feedback path.