# SageMath code for working with number field 47.47.60558628944427886416035618894711378994697503545758179730765967261479053047453845062877188530544503628007178201769.1

# Some of these functions may take a long time to execute (this depends on the field).

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^47 - x^46 - 138*x^45 + 315*x^44 + 8338*x^43 - 29804*x^42 - 276833*x^41 + 1433626*x^40 + 5033859*x^39 - 41190458*x^38 - 30657314*x^37 + 748097961*x^36 - 742659788*x^35 - 8506344013*x^34 + 21519259357*x^33 + 52948548811*x^32 - 268879641855*x^31 - 30332528938*x^30 + 1920252236103*x^29 - 2430736233424*x^28 - 7367030656288*x^27 + 21401598866455*x^26 + 5373046913681*x^25 - 89240242581627*x^24 + 85735098102709*x^23 + 174332690558567*x^22 - 418760640969237*x^21 + 24016008538438*x^20 + 845143941649693*x^19 - 850093833789498*x^18 - 563502754038610*x^17 + 1652337279635119*x^16 - 688139958495907*x^15 - 1132046847057208*x^14 + 1399925903803443*x^13 - 129366579867529*x^12 - 739733924950208*x^11 + 464611724348759*x^10 + 61382685657965*x^9 - 169596960618209*x^8 + 46703998783537*x^7 + 17972540150505*x^6 - 11555848466611*x^5 + 591153994800*x^4 + 757301830397*x^3 - 134933083169*x^2 - 1756074840*x + 132954859)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Autmorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Unit group: 
UK = K.unit_group()

# Unit rank: 
UK.rank()

# Generator for roots of unity: 
UK.torsion_generator()

# Fundamental units: 
UK.fundamental_units()

# Regulator: 
K.regulator()

# Analytic class number formula: 
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ);  K.<a> = NumberField(x^47 - x^46 - 138*x^45 + 315*x^44 + 8338*x^43 - 29804*x^42 - 276833*x^41 + 1433626*x^40 + 5033859*x^39 - 41190458*x^38 - 30657314*x^37 + 748097961*x^36 - 742659788*x^35 - 8506344013*x^34 + 21519259357*x^33 + 52948548811*x^32 - 268879641855*x^31 - 30332528938*x^30 + 1920252236103*x^29 - 2430736233424*x^28 - 7367030656288*x^27 + 21401598866455*x^26 + 5373046913681*x^25 - 89240242581627*x^24 + 85735098102709*x^23 + 174332690558567*x^22 - 418760640969237*x^21 + 24016008538438*x^20 + 845143941649693*x^19 - 850093833789498*x^18 - 563502754038610*x^17 + 1652337279635119*x^16 - 688139958495907*x^15 - 1132046847057208*x^14 + 1399925903803443*x^13 - 129366579867529*x^12 - 739733924950208*x^11 + 464611724348759*x^10 + 61382685657965*x^9 - 169596960618209*x^8 + 46703998783537*x^7 + 17972540150505*x^6 - 11555848466611*x^5 + 591153994800*x^4 + 757301830397*x^3 - 134933083169*x^2 - 1756074840*x + 132954859)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator();  RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))  

# Intermediate fields: 
K.subfields()[1:-1]

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]