# SageMath code for working with number field 38.0.2394510171790650820123124474406353844872595054967993341776661644110188322971389918926750746438903.1

# 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 + 118*x^36 - 435*x^35 + 6131*x^34 - 35235*x^33 + 222008*x^32 - 1286466*x^31 + 5995377*x^30 - 26936649*x^29 + 103513172*x^28 - 328397860*x^27 + 831795745*x^26 - 841346468*x^25 - 6703672958*x^24 + 58117415411*x^23 - 303731983764*x^22 + 1218275129036*x^21 - 3767355799065*x^20 + 7961894667643*x^19 - 2218796696910*x^18 - 81504663492211*x^17 + 502603614584717*x^16 - 2027971305796085*x^15 + 6509260330141933*x^14 - 17551631941921428*x^13 + 40524896388301467*x^12 - 80503012777500047*x^11 + 136388077021556601*x^10 - 193370722349898757*x^9 + 226309207994588218*x^8 - 220960831577107827*x^7 + 187038155057537519*x^6 - 141423416677434751*x^5 + 93202974532409581*x^4 - 51592635735804780*x^3 + 25439606449082092*x^2 - 11768916674310345*x + 3447647463127489)

# 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 + 118*x^36 - 435*x^35 + 6131*x^34 - 35235*x^33 + 222008*x^32 - 1286466*x^31 + 5995377*x^30 - 26936649*x^29 + 103513172*x^28 - 328397860*x^27 + 831795745*x^26 - 841346468*x^25 - 6703672958*x^24 + 58117415411*x^23 - 303731983764*x^22 + 1218275129036*x^21 - 3767355799065*x^20 + 7961894667643*x^19 - 2218796696910*x^18 - 81504663492211*x^17 + 502603614584717*x^16 - 2027971305796085*x^15 + 6509260330141933*x^14 - 17551631941921428*x^13 + 40524896388301467*x^12 - 80503012777500047*x^11 + 136388077021556601*x^10 - 193370722349898757*x^9 + 226309207994588218*x^8 - 220960831577107827*x^7 + 187038155057537519*x^6 - 141423416677434751*x^5 + 93202974532409581*x^4 - 51592635735804780*x^3 + 25439606449082092*x^2 - 11768916674310345*x + 3447647463127489)
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)]