# SageMath code for working with number field 38.0.10517653274833793910788391650485470831673313848992580618621786800117331392788597759645606697493139.1

# 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 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811)

# 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 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811)
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)]