# SageMath code for working with number field 38.0.136635360908492439649635218195335734184282612187547808802428015665989693822130747061222659537193769787.1

# 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 + 171*x^36 - 266*x^35 + 17765*x^34 - 38323*x^33 + 1208419*x^32 - 3128616*x^31 + 60877083*x^30 - 164538138*x^29 + 2281935169*x^28 - 6108816122*x^27 + 65770773122*x^26 - 164067939585*x^25 + 1434043198618*x^24 - 3182597959136*x^23 + 23807715304970*x^22 - 44488778447299*x^21 + 291967423046375*x^20 - 424722852053838*x^19 + 2667130160996125*x^18 - 2974990157993691*x^17 + 18341053997864450*x^16 - 14685757775285777*x^15 + 92760878311304464*x^14 - 59049250104163667*x^13 + 342353164732402796*x^12 - 172475502215832115*x^11 + 890393108976853995*x^10 - 452012764582067042*x^9 + 1563583757111096932*x^8 - 707974172043966504*x^7 + 1816769368682524829*x^6 - 831929210861393479*x^5 + 1134167336866808190*x^4 - 228910989279510535*x^3 + 271437005032832706*x^2 - 30519054068376079*x + 49228485006254761)

# 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 + 171*x^36 - 266*x^35 + 17765*x^34 - 38323*x^33 + 1208419*x^32 - 3128616*x^31 + 60877083*x^30 - 164538138*x^29 + 2281935169*x^28 - 6108816122*x^27 + 65770773122*x^26 - 164067939585*x^25 + 1434043198618*x^24 - 3182597959136*x^23 + 23807715304970*x^22 - 44488778447299*x^21 + 291967423046375*x^20 - 424722852053838*x^19 + 2667130160996125*x^18 - 2974990157993691*x^17 + 18341053997864450*x^16 - 14685757775285777*x^15 + 92760878311304464*x^14 - 59049250104163667*x^13 + 342353164732402796*x^12 - 172475502215832115*x^11 + 890393108976853995*x^10 - 452012764582067042*x^9 + 1563583757111096932*x^8 - 707974172043966504*x^7 + 1816769368682524829*x^6 - 831929210861393479*x^5 + 1134167336866808190*x^4 - 228910989279510535*x^3 + 271437005032832706*x^2 - 30519054068376079*x + 49228485006254761)
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)]