# SageMath code for working with number field 26.26.1919641021107487828653877081110544587155912070949008048128.1

# 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^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)

# 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^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)
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)]