Call the raw CycleExplorer endpoint.
The example will pull eod data from yahoo api first and send to the cycle analsis api to get current dominant cycle:

import requests
import json
import yfinance as yf

#put your cycle.tools API key
cycleAPIkey = 'YOUR_API_KEY'

#set cycle.tools API url endpoint
url = f"https://api.cycle.tools/api/cycles/CycleExplorer?
       dynamicInSampleMethod=true
       &api_key={cycleAPIkey}"

#define the yahoo ticker symbol, e.g. SP 500 = ^GSPC
tickerSymbol = "^GSPC"

#get data on this ticker
tickerData = yf.Ticker(tickerSymbol)

#get the historical prices for this ticker
tickerDf = tickerData.history(interval="1d", period="5y")

#get close data as json doubele array
jsondata = tickerDf['Close'].to_json(orient='records')

#call CycleExplorer endpoint to get current dominant cycle
r= requests.post(url, data=jsondata, 
         headers={'Content-Type':'application/json'})

#get the dominant cycle return data
dominantCycle = r.json()
dominantCycleLength = dominantCycle['length']

#print the current dominant cycle length
print(f"Dominant Cycle Length: {dominantCycleLength}")

#print the json the results
print()
print(json.dumps(r.json(), indent=4, sort_keys=True))

Example output looks like:

Dominant Cycle Length: 165.0

{
 "amplitude": 146.00174004966559,
 "analysisEndDate": "2020-05-03T17:02:20.6771012+00:00",
 "analysisStartDate": "0001-01-01T00:00:00",
 "barsAvailable": 0,
 "barsused": 750,
 "currentPrice": 2830.71,
 "cycleProfitability": 0.75,
 "lastlow": -5.6668009766674174,
 "lasttop": -88.16680097666742,
 "length": 165.0,
 "license": "api.cycle.tools :You agree to the API usage terms and conditions from host ",
 "minbaroffset": 170.66680097666742,
 "nextlow": 159.33319902333258,
 "nexttop": 76.83319902333258,
 "phase": -1.3169251392646713,
 "phase_score": -100,
 "phase_status": "BOTTOM_Departure",
 "phasingScore": 135,
 "statusCode": null,
 "symbol": "manual",
 "timeSeries": []
}

Call the MarketCycles endpoint:
The example will use the integrated crypto datafeed to get the current dominant cycle in Bitcoin.

import requests
import json

#put your cycle.tools API key
cycleAPIkey = 'YOUR_API_KEY'
symbol="BTC" # Bitcoin
market="CDS" # Crypto

#set cycle.tools API url endpoint
url = f"https://api.cycle.tools/api/cycles/MarketCycles/{symbol}?marketType={market}&api_key={cycleAPIkey}"

#call MarketCycles endpoint to get current dominant cycle
r= requests.get(url)

#get the dominant cycle return data
marketCycle = r.json()
marketCycleLength = marketCycle['length']

#print the current dominant cycle length
print(f"Dominant Cycle Length: {marketCycleLength}")

#print the json the results
print()
print(json.dumps(r.json(), indent=4, sort_keys=True))

Example output:

Dominant Cycle Length: 39.0

{
    "amplitude": 418.6291962393411,
    "analysisEndDate": "2020-05-01T00:00:00",
    "analysisStartDate": "2016-11-28T00:00:00Z",
    "barsAvailable": 750,
    "barsused": 750,
    "currentPrice": 8728.58,
    "cycleProfitability": 0.7105263157894737,
    "lastlow": -7.758826601781372,
    "lasttop": -27.258826601781372,
    "length": 39.0,
    "license": "****** : You agree to the API usage terms and conditions from api.cycle.tools",
    "minbaroffset": 46.75882660178137,
    "nextlow": 31.241173398218628,
    "nexttop": 11.741173398218628,
    "phase": -0.15968528545011673,
    "phase_score": 40,
    "phase_status": "Uptrend_Neutral",
    "phasingScore": 0,
    "statusCode": "OK",
    "symbol": "BTC",
    "timeSeries": []
}