A hot mini-Neptune and a temperate, highly eccentric sub-Saturn around the bright K-dwarf TOI-2134

We present the characterisation of an inner mini-Neptune in a 9.2292005$\pm$0.0000063 day orbit and an outer mono-transiting sub-Saturn planet in a 95.50$^{+0.36}_{-0.25}$ day orbit around the moderately active, bright (mv=8.9 mag) K5V star TOI-2134. Based on our analysis of five sectors of TESS data, we determine the radii of TOI-2134b and c to be 2.69$\pm$0.16 R$_{e}$ for the inner planet and 7.27$\pm$0.42 R$_{e}$ for the outer one. We acquired 111 radial-velocity spectra with HARPS-N and 108 radial-velocity spectra with SOPHIE. After careful periodogram analysis, we derive masses for both planets via Gaussian Process regression: 9.13$^{+0.78}_{-0.76}$ M$_{e}$ for TOI-2134b and 41.86$^{+7.69}_{-7.83}$ M$_{e}$ for TOI-2134c. We analysed the photometric and radial-velocity data first separately, then jointly. The inner planet is a mini-Neptune with density consistent with either a water-world or a rocky core planet with a low-mass H/He envelope. The outer planet has a bulk density similar to Saturn's. The outer planet is derived to have a significant eccentricity of 0.67$^{+0.05}_{-0.06}$ from a combination of photometry and RVs. We compute the irradiation of TOI-2134c as 1.45$\pm$0.10 times the bolometric flux received by Earth, positioning it for part of its orbit in the habitable sone of its system. We recommend further RV observations to fully constrain the orbit of TOI-2134c. With an expected Rossiter-McLaughlin (RM) effect amplitude of 7.2$\pm$1.3 m/s, we recommend TOI-2134c for follow-up RM analysis to study the spin-orbit architecture of the system. We calculate the Transmission Spectroscopy Metric, and both planets are suitable for bright-mode NIRCam atmospheric characterisation.