There are multiple genomes in the genome database for organisms especially ones that are commonly used. e.g. NCBI genome database has 23,186 genomes available as of today for Escherichia coli. (LINK)
When working with a particular organism using a genome from RefSeq genomes database is likely your best option since those are manually curated stable genomes. Here is a representative genome for this organism.
$ esearch -db taxonomy -query "1352 [taxID]" | elink -target assembly | efetch -format docsum | xtract -pattern DocumentSummary -element RefSeq,Organism,RefSeq_category,FtpPath_RefSeq | grep -v "na"
GCF_010120755.1 Enterococcus faecium (firmicutes) representative genome ftp://ftp.ncbi.nlm.nih.gov/genomes/all/GCF/010/120/755/GCF_010120755.1_ASM1012075v1
You can only get the name of the organism from a particular SRA dataset but not a specific genome associated with it. You can find the TaxID
in the query below that can be used to find the RefSeq genome as demonstrated above.
$ esearch -db sra -query "ERR1036032" | efetch -format runinfo
Run,ReleaseDate,LoadDate,spots,bases,spots_with_mates,avgLength,size_MB,AssemblyName,download_path,Experiment,LibraryName,LibraryStrategy,LibrarySelection,LibrarySource,LibraryLayout,InsertSize,InsertDev,Platform,Model,SRAStudy,BioProject,Study_Pubmed_id,ProjectID,Sample,BioSample,SampleType,TaxID,ScientificName,SampleName,g1k_pop_code,source,g1k_analysis_group,Subject_ID,Sex,Disease,Tumor,Affection_Status,Analyte_Type,Histological_Type,Body_Site,CenterName,Submission,dbgap_study_accession,Consent,RunHash,ReadHash
ERR1036032,2015-09-29 05:07:35,2016-03-04 14:45:45,2128045,425609000,2128045,200,183,ASM25094v1,https://sra-downloadb.st-va.ncbi.nlm.nih.gov/sos1/sra-pub-run-8/ERR1036032/ERR1036032.1,ERX1114766,13764563,WGS,RANDOM,GENOMIC,PAIRED,500,0,ILLUMINA,Illumina HiSeq 2000,ERP009805,PRJEB8769,,293294,ERS683353,SAMEA3304528,simple,1352,Enterococcus faecium,SAMEA3304528,,,,,,,no,,,,,THE WELLCOME TRUST SANGER INSTITUTE,ERA490686,,public,16BA4EEAD61804CF2C7B75E487BEFC64,159B6E74B5596FD71E84956CF20C333B
You can then use NCBI's new tool called datasets
(LINK) to download the genome sequence
$ datasets download genome taxon 1352 --exclude-protein