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Properties

Label 16.8.72504279208...1856.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{66}\cdot 7^{6}\cdot 17^{4}$
Root discriminant $73.50$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32481, 390096, -664992, -2773584, -357176, 700720, 77032, 178864, 13414, -5152, -6408, -3472, -480, -144, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 144*x^13 - 480*x^12 - 3472*x^11 - 6408*x^10 - 5152*x^9 + 13414*x^8 + 178864*x^7 + 77032*x^6 + 700720*x^5 - 357176*x^4 - 2773584*x^3 - 664992*x^2 + 390096*x - 32481)
 
gp: K = bnfinit(x^16 + 16*x^14 - 144*x^13 - 480*x^12 - 3472*x^11 - 6408*x^10 - 5152*x^9 + 13414*x^8 + 178864*x^7 + 77032*x^6 + 700720*x^5 - 357176*x^4 - 2773584*x^3 - 664992*x^2 + 390096*x - 32481, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 144 x^{13} - 480 x^{12} - 3472 x^{11} - 6408 x^{10} - 5152 x^{9} + 13414 x^{8} + 178864 x^{7} + 77032 x^{6} + 700720 x^{5} - 357176 x^{4} - 2773584 x^{3} - 664992 x^{2} + 390096 x - 32481 \)

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:  \(725042792081759922538839801856=2^{66}\cdot 7^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.50$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{459} a^{14} + \frac{2}{51} a^{13} + \frac{142}{459} a^{12} + \frac{25}{51} a^{11} - \frac{73}{153} a^{10} - \frac{97}{459} a^{9} - \frac{5}{17} a^{8} + \frac{149}{459} a^{7} + \frac{139}{459} a^{6} - \frac{65}{459} a^{5} + \frac{145}{459} a^{4} + \frac{160}{459} a^{3} - \frac{200}{459} a^{2} + \frac{8}{17} a - \frac{2}{51}$, $\frac{1}{851689332213561292086450355719685397078989485297} a^{15} - \frac{6848145742960402743692878966935178112242823}{283896444071187097362150118573228465692996495099} a^{14} + \frac{97825688997412612085318402919617507507827352968}{851689332213561292086450355719685397078989485297} a^{13} + \frac{60170984141847996384640979170099930472889144760}{283896444071187097362150118573228465692996495099} a^{12} + \frac{6551428697111994510129024701920723120897204924}{16699790827716888080126477563131086217235087947} a^{11} - \frac{67871545298106649723669891575770992563453902104}{851689332213561292086450355719685397078989485297} a^{10} - \frac{69727357513492863214988830568200140945651687877}{283896444071187097362150118573228465692996495099} a^{9} - \frac{321271485287018124532909322412464116500807556072}{851689332213561292086450355719685397078989485297} a^{8} + \frac{49790912909348558695731660905602454153843829850}{851689332213561292086450355719685397078989485297} a^{7} + \frac{33509262120050192945316889195586456810264621089}{851689332213561292086450355719685397078989485297} a^{6} - \frac{415269750404833078371028768726387050427575170585}{851689332213561292086450355719685397078989485297} a^{5} - \frac{154135021499276775880762181257552109465092407896}{851689332213561292086450355719685397078989485297} a^{4} - \frac{253134545509586809523296840881616381291506208181}{851689332213561292086450355719685397078989485297} a^{3} - \frac{113641769613262824213113097938633889952120679501}{283896444071187097362150118573228465692996495099} a^{2} - \frac{17783676177390343318604433049125325369260102092}{94632148023729032454050039524409488564332165033} a - \frac{11004130614478177639520588595210149274726047291}{31544049341243010818016679841469829521444055011}$

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:  \( 1454683984.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14336.1, \(\Q(\zeta_{16})^+\), 4.4.7168.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$