# SageMath code for working with number field 38.38.2907625224541948887418190194159474648332884150922805055774449275194637293139188364462591928677.1

# 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^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451)

# 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^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451)
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)]