# SageMath code for working with number field 35.35.73261800077965937220382205398471606200231960977600836588587975360331959691780081.1

# 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^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503)

# 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^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503)
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)]