# SageMath code for working with number field 21.9.834507966212271820403374346664412679514757637054312831716289049278398513152.1.
# 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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Automorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Narrow class group: 
K.narrow_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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)
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()

# Frobenius cycle types: 
# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]