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I have a star that I need to calculate the absolute magnitude of. I am given the temperature, luminosity, radius, mass, and distance in light-years. So I am wondering, what is the formula to compute this, using only these terms?

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Yes. You derive the absolute bolometric magnitude by reference to the Sun. The Sun has a bolometric magnitude (i.e. integrated over all wavelengths) of about 4.75. Thus the difference in bolometric magnitudes is given by the log of the ratio of luminosities like so $$ M_{\rm bol} = 4.75 -2.5 \log_{10} \left(\frac{L_*}{L_{\odot}}\right)\ ,$$ where $L_{\odot} = 3.8\times 10^{26}$ W is the luminosity of the Sun and $L_*$ is the luminosity of the star.

To get the absolute visual magnitude then you need to subtract a bolometric correction, which to first order, depends on the temperature of the star (it also depends to second order on the composition and gravity). $$ M_V = M_{\rm bol} - BC_V\ .$$

A table of bolometric corrections vs temperature can be found in the 6th column of this authoritative table, which is in common use by astronomers/astrophysicists.

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You just use the Luminosity:

$M = -2.5 \log_{10}(L / L_0)$

Where $L_0$ is the luminosity of a magnitude 0 star $L_0=3\times10^{28}\mathrm{W}=79L_\odot$

You don't need the other variables. You would need the distance if you wanted to calculate the apparent magnitude. You could also use the radius and mass to estimate the luminosity, if you weren't given it.

This gives the bolometric magnitude, which includes the brightness of the star at infrared and ultraviolet wavelengths (which, of course, we can't see) You could use the temperature to correct for this and get a visual magnitude, but the calculation then becomes more complex, you would need to estimate the temperature from the mass and radius, which is left as an exercise for the reader.

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