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Properties

Label 16.8.39728624488...6209.7
Degree $16$
Signature $[8, 4]$
Discriminant $71^{10}\cdot 73^{14}$
Root discriminant $612.98$
Ramified primes $71, 73$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7578526976, -11718369616, 3843574804, 5009780308, -1833365633, -891092496, 282455831, 77486790, -21733511, -2562342, 746968, -14574, -3913, 1374, -179, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 179*x^14 + 1374*x^13 - 3913*x^12 - 14574*x^11 + 746968*x^10 - 2562342*x^9 - 21733511*x^8 + 77486790*x^7 + 282455831*x^6 - 891092496*x^5 - 1833365633*x^4 + 5009780308*x^3 + 3843574804*x^2 - 11718369616*x + 7578526976)
 
gp: K = bnfinit(x^16 - 4*x^15 - 179*x^14 + 1374*x^13 - 3913*x^12 - 14574*x^11 + 746968*x^10 - 2562342*x^9 - 21733511*x^8 + 77486790*x^7 + 282455831*x^6 - 891092496*x^5 - 1833365633*x^4 + 5009780308*x^3 + 3843574804*x^2 - 11718369616*x + 7578526976, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 179 x^{14} + 1374 x^{13} - 3913 x^{12} - 14574 x^{11} + 746968 x^{10} - 2562342 x^{9} - 21733511 x^{8} + 77486790 x^{7} + 282455831 x^{6} - 891092496 x^{5} - 1833365633 x^{4} + 5009780308 x^{3} + 3843574804 x^{2} - 11718369616 x + 7578526976 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(397286244885795491769071181517306466749106209=71^{10}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.98$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $71, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{5}{16} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{96} a^{12} + \frac{1}{48} a^{11} + \frac{1}{96} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} + \frac{1}{16} a^{5} - \frac{5}{32} a^{4} - \frac{3}{8} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} - \frac{5}{192} a^{11} + \frac{7}{192} a^{10} + \frac{1}{96} a^{9} + \frac{1}{32} a^{8} + \frac{5}{64} a^{7} + \frac{1}{64} a^{6} - \frac{3}{64} a^{5} + \frac{15}{64} a^{4} - \frac{13}{48} a^{3} + \frac{3}{16} a^{2} + \frac{1}{12} a$, $\frac{1}{1545432576} a^{14} + \frac{67827}{257572096} a^{13} - \frac{13139}{48294768} a^{12} + \frac{793469}{128786048} a^{11} - \frac{67291777}{1545432576} a^{10} - \frac{2863579}{48294768} a^{9} + \frac{140497297}{1545432576} a^{8} + \frac{3988289}{32196512} a^{7} - \frac{3167521}{32196512} a^{6} + \frac{61531897}{257572096} a^{5} + \frac{271878827}{1545432576} a^{4} - \frac{8670265}{386358144} a^{3} + \frac{105954389}{386358144} a^{2} + \frac{8222779}{96589536} a + \frac{2912723}{6036846}$, $\frac{1}{77960935088513058815368077720596635225168716307021824} a^{15} + \frac{6002511582092550719519426226734217385398407}{77960935088513058815368077720596635225168716307021824} a^{14} - \frac{34091717755681099656114680464539189878079715676339}{38980467544256529407684038860298317612584358153510912} a^{13} + \frac{29054133027186740193829223525098825099403828963477}{6496744590709421567947339810049719602097393025585152} a^{12} + \frac{699593437692998962826876786204507771963591755402201}{25986978362837686271789359240198878408389572102340608} a^{11} - \frac{3889224672380404051373836424345372482942822252442741}{77960935088513058815368077720596635225168716307021824} a^{10} - \frac{619013580123449153706187776494656624567598101992367}{77960935088513058815368077720596635225168716307021824} a^{9} + \frac{800519279366375897864653347463276533275198300338759}{25986978362837686271789359240198878408389572102340608} a^{8} + \frac{17207812156700724944920054345105339467779668999091}{812093073838677695993417476256214950262174128198144} a^{7} + \frac{84900588355608813777768936713920033206669280691761}{12993489181418843135894679620099439204194786051170304} a^{6} - \frac{682459757207322308903976840134313097023912872750407}{77960935088513058815368077720596635225168716307021824} a^{5} - \frac{4894877984622826813007242307255149903288095462944735}{25986978362837686271789359240198878408389572102340608} a^{4} + \frac{129685394575026948106218413001703729576109249460607}{406046536919338847996708738128107475131087064099072} a^{3} + \frac{1564366436499530165451478233736345232068751190390231}{6496744590709421567947339810049719602097393025585152} a^{2} - \frac{2269802133276509679262181726183204965355150937842417}{4872558443032066175960504857537289701573044769188864} a - \frac{27813649138153702714321804868069507277618412595205}{101511634229834711999177184532026868782771766024768}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4327900930910000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.280732967047165372057.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
71Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$