Determining the exact conception date can be challenging, especially with irregular periods. However, there are a few methods you can use to estimate it. One way is by counting back from the first day of your last period, assuming a typical menstrual cycle of 28 days. In your case, if your ultrasound suggests that your last period was on December 9th and your cycle is usually around 28 days long, it is likely that conception occurred around December 23rd or shortly after.
Another method is to track changes in cervical mucus consistency and basal body temperature (BBT). Cervical mucus becomes thin and slippery during ovulation, indicating fertility. BBT rises slightly after ovulation occurs. By monitoring these changes over time, you may be able to pinpoint when ovulation and therefore conception took place.
However, keep in mind that these methods provide estimates rather than precise dates. It's always best to consult with a healthcare professional who can evaluate your individual circumstances and provide more accurate information based on additional factors such as hormone levels or other diagnostic tests.