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Properties

Label 16.8.54620748506...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}$
Root discriminant $46.89$
Ramified primes $2, 5, 11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73961, -223458, 119082, 181342, -228247, 55282, 61900, -53100, 17936, -630, -3040, 2002, -587, 42, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 22*x^14 + 42*x^13 - 587*x^12 + 2002*x^11 - 3040*x^10 - 630*x^9 + 17936*x^8 - 53100*x^7 + 61900*x^6 + 55282*x^5 - 228247*x^4 + 181342*x^3 + 119082*x^2 - 223458*x + 73961)
 
gp: K = bnfinit(x^16 - 8*x^15 + 22*x^14 + 42*x^13 - 587*x^12 + 2002*x^11 - 3040*x^10 - 630*x^9 + 17936*x^8 - 53100*x^7 + 61900*x^6 + 55282*x^5 - 228247*x^4 + 181342*x^3 + 119082*x^2 - 223458*x + 73961, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 22 x^{14} + 42 x^{13} - 587 x^{12} + 2002 x^{11} - 3040 x^{10} - 630 x^{9} + 17936 x^{8} - 53100 x^{7} + 61900 x^{6} + 55282 x^{5} - 228247 x^{4} + 181342 x^{3} + 119082 x^{2} - 223458 x + 73961 \)

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:  \(546207485068843417600000000=2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.89$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{2}{25} a^{9} + \frac{1}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{11}{25} a^{3} - \frac{12}{25} a^{2} - \frac{1}{25} a - \frac{2}{25}$, $\frac{1}{550} a^{12} - \frac{3}{275} a^{11} - \frac{4}{275} a^{10} - \frac{7}{275} a^{9} + \frac{17}{275} a^{8} - \frac{24}{275} a^{7} - \frac{2}{55} a^{6} - \frac{7}{55} a^{5} - \frac{52}{275} a^{4} + \frac{4}{275} a^{3} - \frac{38}{275} a^{2} - \frac{81}{275} a - \frac{47}{110}$, $\frac{1}{550} a^{13} + \frac{2}{275} a^{10} + \frac{19}{275} a^{9} - \frac{2}{55} a^{8} - \frac{1}{25} a^{7} + \frac{4}{275} a^{6} + \frac{13}{275} a^{5} - \frac{3}{25} a^{4} + \frac{118}{275} a^{3} + \frac{87}{275} a^{2} - \frac{151}{550} a - \frac{89}{275}$, $\frac{1}{30250} a^{14} + \frac{2}{15125} a^{13} - \frac{21}{30250} a^{12} - \frac{287}{15125} a^{11} + \frac{243}{15125} a^{10} - \frac{216}{15125} a^{9} + \frac{8}{121} a^{8} + \frac{1091}{15125} a^{7} + \frac{127}{3025} a^{6} - \frac{6286}{15125} a^{5} + \frac{603}{1375} a^{4} - \frac{1362}{15125} a^{3} - \frac{4041}{30250} a^{2} + \frac{5842}{15125} a - \frac{14279}{30250}$, $\frac{1}{11943188692248752049262250} a^{15} + \frac{1535857374524507257}{2388637738449750409852450} a^{14} - \frac{2399906821054759841346}{5971594346124376024631125} a^{13} + \frac{22583921191109319749}{2388637738449750409852450} a^{12} - \frac{64308039365726922105009}{5971594346124376024631125} a^{11} - \frac{64752226338709167633283}{5971594346124376024631125} a^{10} + \frac{411571338462076827864994}{5971594346124376024631125} a^{9} - \frac{76443156875847782554664}{5971594346124376024631125} a^{8} - \frac{61020067281870632924409}{5971594346124376024631125} a^{7} + \frac{537792833471606048949729}{5971594346124376024631125} a^{6} - \frac{6525800815238182312127}{28572221751791272845125} a^{5} + \frac{1320321249762954090022681}{5971594346124376024631125} a^{4} - \frac{1141453341771929083068647}{2388637738449750409852450} a^{3} - \frac{4653736736504656807131607}{11943188692248752049262250} a^{2} - \frac{523258549082856025380798}{1194318869224875204926225} a - \frac{133497405053938868359569}{1085744426568068368114750}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 38808361.0147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5225.1, 4.4.4400.1, 4.4.7600.1, 8.8.6988960000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. 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
$2$2.8.12.24$x^{8} + 4 x^{6} + 28 x^{4} + 80$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.12.22$x^{8} + 4 x^{7} + 16 x^{3} + 48$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$