Solar activities directly or indirectly affect space missions, geophysical environment, space climate, and human activities. We used the long short-term memory (LSTM) deep learning method to predict the amplitude and peak time of solar cycles (SCs) 25 and 26 by using the monthly relative sunspot number data taken from the National Astronomical Observatory of Japan (NAOJ). The dataset is divided into eight schemes of two to nine slices for training, showing that the five-slice LSTM model with root mean square error of 11.38 is the optimal model. According to the prediction, SC 25 will be about 21% stronger than SC 24, with a peak of 135.2 occurring in 2024 April. SC 26 will be similar to SC 25 and reach its peak of 135.0 in 2035 January. Our analysis results indicate that the sunspot data from NAOJ is highly credible and comparable.