We present the analysis of a total of 177h of high-quality optical time-series photometry of the helium atmosphere pulsating white dwarf (DBV) EC 20058-5234. The bulk of the observations (135h) were obtained during a WET campaign (XCOV15) in July 1997 that featured coordinated observing from 4 southern observatory sites over an 8-day period. The remaining data (42h) were obtained in June 2004 at Mt John Observatory in NZ over a one-week observing period. This work significantly extends the discovery observations of this low-amplitude (few percent) pulsator by increasing the number of detected frequencies from 8 to 18, and employs a simulation procedure to confirm the reality of these frequencies to a high level of significance (1 in 1000). The nature of the observed pulsation spectrum precludes identification of unique pulsation mode properties using any clearly discernable trends. However, we have used a global modelling procedure employing genetic algorithm techniques to identify the n, l values of 8 pulsation modes, and thereby obtain asteroseismic measurements of several model parameters, including the stellar mass (0.55 M_sun) and T_eff (~28200 K). These values are consistent with those derived from published spectral fitting: T_eff ~ 28400 K and log g ~ 7.86. We also present persuasive evidence from apparent rotational mode splitting for two of the modes that indicates this compact object is a relatively rapid rotator with a period of 2h. In direct analogy with the corresponding properties of the hydrogen (DAV) atmosphere pulsators, the stable low-amplitude pulsation behaviour of EC 20058 is entirely consistent with its inferred effective temperature, which indicates it is close to the blue edge of the DBV instability strip. (abridged)
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