  We construct the Type Ia supernovae (SNe Ia) luminosity function (LF) using
the Zwicky Transient Facility Bright Transient Survey (BTS) catalogue. While
this magnitude-limited survey has an unprecedented number of objects, it
suffers from large distance uncertainties and lacks an estimation of host
extinction. We bypass these issues by calculating the intrinsic luminosities
from the shape parameters of the light curve's $ g $ and $ r $ bands, with the
luminosities calibrated from the well observed SNe Ia sample of the Carnegie
Supernova Project, allowing us to construct, for the first time, the intrinsic
LF of SNe Ia. We then use a novel tight relation between the color stretch and
the synthesized $^{56}$Ni mass, $M_\mathrm{Ni56}$, to determine the
$M_\mathrm{Ni56}$ distribution of SNe Ia. We find that the LFs are unimodal,
with their peaks in line with previous results, but have a much lower rate of
dim events and luminous events. We show that the features on top of the
unimodal LF-derived distributions are all compatible with statistical noise,
consistent with a single progenitor channel for the explosions. We further
derive, for the first time, the SNe Ia distribution of host galaxy extinction,
and find a mean selective extinction of $E(B-V)\approx0.1$ and a non-negligible
fraction with large, $ >1\,\text{mag} $, extinction in the optical bands. The
high extinction is typical for luminous SNe, supporting their young population
origin.
