# SageMath code for working with number field 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1

# 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^42 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069)

# 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^42 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069)
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)]