# SageMath code for working with number field 47.47.6807739517051283990370106856968124192474229146184900948675688481754810058334971205734548194436392534997735758198660848138706901604387352879006731328411841.1

# 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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449)

# 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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449)
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)]