# SageMath code for working with number field 35.35.42664569157776260194145291158312909027607801510075223187489424993547224224125412187641.1

# 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 - 2*x^34 - 147*x^33 + 384*x^32 + 8999*x^31 - 28384*x^30 - 300934*x^29 + 1116603*x^28 + 6027458*x^27 - 26518095*x^26 - 73782007*x^25 + 404019590*x^24 + 521840181*x^23 - 4069268247*x^22 - 1503940067*x^21 + 27441438677*x^20 - 6471979421*x^19 - 124171354119*x^18 + 81524582969*x^17 + 375114725905*x^16 - 364687984188*x^15 - 747339096335*x^14 + 926308497451*x^13 + 961292881275*x^12 - 1451786049968*x^11 - 771321612374*x^10 + 1427632011474*x^9 + 365928432486*x^8 - 868346663001*x^7 - 97103750919*x^6 + 312022190475*x^5 + 17703737306*x^4 - 59996070990*x^3 - 4579250401*x^2 + 4824514261*x + 750695467)

# 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 - 2*x^34 - 147*x^33 + 384*x^32 + 8999*x^31 - 28384*x^30 - 300934*x^29 + 1116603*x^28 + 6027458*x^27 - 26518095*x^26 - 73782007*x^25 + 404019590*x^24 + 521840181*x^23 - 4069268247*x^22 - 1503940067*x^21 + 27441438677*x^20 - 6471979421*x^19 - 124171354119*x^18 + 81524582969*x^17 + 375114725905*x^16 - 364687984188*x^15 - 747339096335*x^14 + 926308497451*x^13 + 961292881275*x^12 - 1451786049968*x^11 - 771321612374*x^10 + 1427632011474*x^9 + 365928432486*x^8 - 868346663001*x^7 - 97103750919*x^6 + 312022190475*x^5 + 17703737306*x^4 - 59996070990*x^3 - 4579250401*x^2 + 4824514261*x + 750695467)
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)]