# SageMath code for working with number field 38.0.15223168714879313546692095257379448420591016496978036941227483116202289492875331751113046747.1
# Some of these functions may take a long time to execute (this depends on the field).
# Define the number field:
x = polygen(QQ); K. = NumberField(x^38 - x^37 + 91*x^36 - 24*x^35 + 5113*x^34 - 166*x^33 + 173717*x^32 + 27194*x^31 + 4239797*x^30 + 1325317*x^29 + 74027272*x^28 + 40999344*x^27 + 983326589*x^26 + 743120555*x^25 + 10059741791*x^24 + 9563481092*x^23 + 81070213075*x^22 + 87547591746*x^21 + 511882174610*x^20 + 601337789804*x^19 + 2539884758984*x^18 + 3042827513863*x^17 + 9675050790560*x^16 + 11543123809355*x^15 + 28268796522511*x^14 + 31776668507252*x^13 + 60600608962241*x^12 + 63076854532617*x^11 + 95016529831875*x^10 + 85766481354393*x^9 + 98440678914956*x^8 + 74640315704381*x^7 + 67352150639088*x^6 + 40110464366364*x^5 + 23287561297245*x^4 + 7350839404496*x^3 + 1810153278801*x^2 + 182761486103*x + 13841287201)
# 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. = NumberField(x^38 - x^37 + 91*x^36 - 24*x^35 + 5113*x^34 - 166*x^33 + 173717*x^32 + 27194*x^31 + 4239797*x^30 + 1325317*x^29 + 74027272*x^28 + 40999344*x^27 + 983326589*x^26 + 743120555*x^25 + 10059741791*x^24 + 9563481092*x^23 + 81070213075*x^22 + 87547591746*x^21 + 511882174610*x^20 + 601337789804*x^19 + 2539884758984*x^18 + 3042827513863*x^17 + 9675050790560*x^16 + 11543123809355*x^15 + 28268796522511*x^14 + 31776668507252*x^13 + 60600608962241*x^12 + 63076854532617*x^11 + 95016529831875*x^10 + 85766481354393*x^9 + 98440678914956*x^8 + 74640315704381*x^7 + 67352150639088*x^6 + 40110464366364*x^5 + 23287561297245*x^4 + 7350839404496*x^3 + 1810153278801*x^2 + 182761486103*x + 13841287201)
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)]