# SageMath code for working with number field 45.45.2524373658125154706394844918628378076658288807428864227492811533944564375098081397291186234112097170769516929.1

# 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^45 - 12*x^44 - 80*x^43 + 1424*x^42 + 1516*x^41 - 74554*x^40 + 63158*x^39 + 2282501*x^38 - 4255902*x^37 - 45636421*x^36 + 116693559*x^35 + 630121996*x^34 - 1971613347*x^33 - 6193698077*x^32 + 22855343401*x^31 + 44007670585*x^30 - 190617019593*x^29 - 226920195907*x^28 + 1172746429282*x^27 + 842990049201*x^26 - 5394671280584*x^25 - 2206384597765*x^24 + 18666651459717*x^23 + 3871477687999*x^22 - 48596558981735*x^21 - 4036750693451*x^20 + 94693820108958*x^19 + 1546861746446*x^18 - 136570614884744*x^17 + 956863921562*x^16 + 143064103358001*x^15 + 102341290612*x^14 - 105649963477592*x^13 - 2816833651750*x^12 + 52484763220205*x^11 + 2919265367283*x^10 - 16311452643376*x^9 - 1104335997267*x^8 + 2857645920655*x^7 + 105919440856*x^6 - 255094574859*x^5 + 8466426668*x^4 + 8654209646*x^3 - 1026667213*x^2 + 14388119*x + 1507921)

# 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^45 - 12*x^44 - 80*x^43 + 1424*x^42 + 1516*x^41 - 74554*x^40 + 63158*x^39 + 2282501*x^38 - 4255902*x^37 - 45636421*x^36 + 116693559*x^35 + 630121996*x^34 - 1971613347*x^33 - 6193698077*x^32 + 22855343401*x^31 + 44007670585*x^30 - 190617019593*x^29 - 226920195907*x^28 + 1172746429282*x^27 + 842990049201*x^26 - 5394671280584*x^25 - 2206384597765*x^24 + 18666651459717*x^23 + 3871477687999*x^22 - 48596558981735*x^21 - 4036750693451*x^20 + 94693820108958*x^19 + 1546861746446*x^18 - 136570614884744*x^17 + 956863921562*x^16 + 143064103358001*x^15 + 102341290612*x^14 - 105649963477592*x^13 - 2816833651750*x^12 + 52484763220205*x^11 + 2919265367283*x^10 - 16311452643376*x^9 - 1104335997267*x^8 + 2857645920655*x^7 + 105919440856*x^6 - 255094574859*x^5 + 8466426668*x^4 + 8654209646*x^3 - 1026667213*x^2 + 14388119*x + 1507921)
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)]