# SageMath code for working with number field 37.37.1716744492644476569905693744668682750842648214113467507247085096126466572919601.1

# 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^37 - x^36 - 72*x^35 + 67*x^34 + 2278*x^33 - 1964*x^32 - 41849*x^31 + 33262*x^30 + 497142*x^29 - 362207*x^28 - 4027127*x^27 + 2672215*x^26 + 22871279*x^25 - 13719683*x^24 - 92273266*x^23 + 49606380*x^22 + 265291071*x^21 - 126402144*x^20 - 540999671*x^19 + 224635365*x^18 + 773506030*x^17 - 271832880*x^16 - 761221384*x^15 + 214848879*x^14 + 502240682*x^13 - 103923082*x^12 - 213804070*x^11 + 27920639*x^10 + 55105551*x^9 - 3603129*x^8 - 7708785*x^7 + 235974*x^6 + 497852*x^5 - 22640*x^4 - 12819*x^3 + 1020*x^2 + 28*x - 1)

# 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^37 - x^36 - 72*x^35 + 67*x^34 + 2278*x^33 - 1964*x^32 - 41849*x^31 + 33262*x^30 + 497142*x^29 - 362207*x^28 - 4027127*x^27 + 2672215*x^26 + 22871279*x^25 - 13719683*x^24 - 92273266*x^23 + 49606380*x^22 + 265291071*x^21 - 126402144*x^20 - 540999671*x^19 + 224635365*x^18 + 773506030*x^17 - 271832880*x^16 - 761221384*x^15 + 214848879*x^14 + 502240682*x^13 - 103923082*x^12 - 213804070*x^11 + 27920639*x^10 + 55105551*x^9 - 3603129*x^8 - 7708785*x^7 + 235974*x^6 + 497852*x^5 - 22640*x^4 - 12819*x^3 + 1020*x^2 + 28*x - 1)
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)]