# SageMath code for working with number field 35.35.1455622807785591094953547155658149343464416905925495778881639129313848614446169.1

# 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 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859)

# 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 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859)
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)]