  The Ch-type asteroid (130) Elektra is orbited by three moons, making it the
first quadruple system in the main asteroid belt.
  We aim to characterise the irregular shape of Elektra and construct a
complete orbital model of its unique moon system.
  We applied the All-Data Asteroid Modelling (ADAM) algorithm to 60 light
curves of Elektra, including our new measurements, 46 adaptive-optics (AO)
images obtained by the VLT/SPHERE and Keck/Nirc2 instruments, and two stellar
occultation profiles.
  For the orbital model, we used an advanced $N$-body integrator, which
includes a multipole expansion of the central body (with terms up to the order
$\ell = 6$), mutual perturbations, internal tides, as well as the external tide
of the Sun acting on the orbits. We fitted the astrometry measured with respect
to the central body and also relatively, with respect to the moons themselves.
  We obtained a revised shape model of Elektra with the volume-equivalent
diameter $(201\pm 2)\,{\rm km}$. Out of two pole solutions, $(\lambda, \beta) =
(189; -88)\,{\rm deg}$ is preferred, because the other one leads to an
incorrect orbital evolution of the moons. We also identified the true orbital
period of the third moon S/2014 (130) 2 as $P_2 = (1.642112 \pm 0.000400)\,{\rm
d}$, which is in between the other periods, $P_1 \simeq 1.212\,{\rm d}$, $P_3
\simeq 5.300\,{\rm d}$, of S/2014 (130) 1 and S/2003 (130) 1, respectively. The
resulting mass of Elektra, $(6.606 \substack{+0.007 \\ -0.013}) \times
10^{18}\,{\rm kg}$, is precisely constrained by all three orbits. Its bulk
density is then $(1.536 \pm 0.038)\,{\rm g\,cm}^{-3}$. The expansion with the
assumption of homogeneous interior leads to the oblateness $J_2 = -C_{20}
\simeq 0.16$. However, the best-fit precession rates indicate a slightly higher
value, ${\simeq}\,0.18$.
