# SageMath code for working with number field 35.35.4974456597909264757129569420535951139863442014128922856612718940624179015847481515681.1

# 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 - 181*x^33 + 470*x^32 + 14259*x^31 - 45478*x^30 - 638530*x^29 + 2447153*x^28 + 17767052*x^27 - 82345759*x^26 - 312539633*x^25 + 1833247298*x^24 + 3264215879*x^23 - 27715265019*x^22 - 13487454777*x^21 + 286033632049*x^20 - 123264134993*x^19 - 1983204888009*x^18 + 2337315606897*x^17 + 8777927035267*x^16 - 17238219604018*x^15 - 21289728738165*x^14 + 71563023595709*x^13 + 8692436762607*x^12 - 169783075654432*x^11 + 94786315419610*x^10 + 202072200901306*x^9 - 240242794434974*x^8 - 58566297948415*x^7 + 212040077699383*x^6 - 73316281317651*x^5 - 51014137537536*x^4 + 37974482870362*x^3 - 1868368126575*x^2 - 3872034006535*x + 804494744591)

# 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 - 181*x^33 + 470*x^32 + 14259*x^31 - 45478*x^30 - 638530*x^29 + 2447153*x^28 + 17767052*x^27 - 82345759*x^26 - 312539633*x^25 + 1833247298*x^24 + 3264215879*x^23 - 27715265019*x^22 - 13487454777*x^21 + 286033632049*x^20 - 123264134993*x^19 - 1983204888009*x^18 + 2337315606897*x^17 + 8777927035267*x^16 - 17238219604018*x^15 - 21289728738165*x^14 + 71563023595709*x^13 + 8692436762607*x^12 - 169783075654432*x^11 + 94786315419610*x^10 + 202072200901306*x^9 - 240242794434974*x^8 - 58566297948415*x^7 + 212040077699383*x^6 - 73316281317651*x^5 - 51014137537536*x^4 + 37974482870362*x^3 - 1868368126575*x^2 - 3872034006535*x + 804494744591)
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)]