# SageMath code for working with number field 43.43.162686032778208990102858628859785420567496242104134005559503199497609643882923419981647276367075859293620549051195773051892887390454194801.1

# (Note that not all these functions may be available, and some may take a long time to execute.)

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^43 - 903*x^41 - 645*x^40 + 358276*x^39 + 487362*x^38 - 82668446*x^37 - 161379172*x^36 + 12386889944*x^35 + 31035129471*x^34 - 1275092913940*x^33 - 3868686195114*x^32 + 93052767268909*x^31 + 330473143424644*x^30 - 4897798025442393*x^29 - 19967338988758228*x^28 + 187329847291648022*x^27 + 869728040966016241*x^26 - 5201925841676982029*x^25 - 27621776964130665961*x^24 + 103748981858304966660*x^23 + 643326429466437698421*x^22 - 1445787294046167541816*x^21 - 10999588529245966977339*x^20 + 13140703542377826450494*x^19 + 137573117593988043652238*x^18 - 60838518194479293931722*x^17 - 1247516677692517201034941*x^16 - 134666416459554320685647*x^15 + 8070410654031859605939559*x^14 + 4203376282667672263555078*x^13 - 36228120784944419229616219*x^12 - 30515963297503545387850753*x^11 + 107534691527795878017440861*x^10 + 119306453339061961678830425*x^9 - 192818992256047972290970619*x^8 - 263326021640166500214931210*x^7 + 170563929616560831037152772*x^6 + 294222510747058121054570076*x^5 - 33622183448420036732904682*x^4 - 118832831442545361067982885*x^3 - 7933826886293481520543865*x^2 + 14571491290875996179048369*x + 2572343484535669027372727)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

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

# 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()

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
p = 7;  # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
[(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]