If you want an E-string based chord shape, fret the root, the fifth, the octave, and then either barre the G string down from the root for the minor, or fret one note higher on the G string for the major. These are the E-shape barre chords. Ignore the two highest strings here:
These can be quite a stretch on bass though, especially below 5th fret, and as I mentioned, I don’t really think E-string rooted chords sound very good on bass. A and D string rooted power chords sound way better IMO.
I actually am generally not a fan of anything but power chords on bass. Most others get real muddy fast IMO. Really prefer guitar for this.
But this still begs the question of if the OP was asking about chords or chord tones/arpeggios 