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This altitude does not have to be the altitude of first possible visibility.
Throughout the year the star will rise approximately four minutes earlier each successive sunrise.
This observation was almost certainly made at Itj-Tawy, the Twelfth Dynasty capital, which would date the Twelfth Dynasty from 1963 to 1786 The Sothic cycle is a specific example of two cycles of differing length interacting to cycle together, here called a tertiary cycle.
This is mathematically defined by the formula 1/a 1/b = 1/t or half the harmonic mean.
One also needs to know the place of observation, since the latitude of the observation changes the day when the heliacal rising of Sirius occurs, and mislocating an observation can potentially change the resulting chronology by several decades.
Three specific observations of the heliacal rise of Sirius are extremely important for Egyptian chronology.
A heliacal rise of Sirius was recorded by Censorinus as having happened on the Egyptian New Year's Day between 140-142.
This value would drop to about 1456 calendar years by the Middle Kingdom.
This cycle was first noticed by Eduard Meyer in 1904, who then carefully combed known Egyptian inscriptions and written materials to find any mention of the calendar dates when Sirius rose at dawn.
He found six of them, on which the dates of much of the conventional Egyptian chronology are based.
The axis of the earth wobbles around slowly moving the observer and changing the observation of the event.
If the axis swings the observer closer to the event its observational year will be shortened.