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Properties

Label 16.0.74015667246...1393.5
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{15}$
Root discriminant $477.90$
Ramified primes $43, 97$
Class number $256$ (GRH)
Class group $[8, 32]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![162278712581, -19356775495, 19413309680, -132257608, 119367930, -78549692, 50381510, -20283562, 4546826, 274360, 20192, -41022, 6928, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 6928*x^12 - 41022*x^11 + 20192*x^10 + 274360*x^9 + 4546826*x^8 - 20283562*x^7 + 50381510*x^6 - 78549692*x^5 + 119367930*x^4 - 132257608*x^3 + 19413309680*x^2 - 19356775495*x + 162278712581)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 6928*x^12 - 41022*x^11 + 20192*x^10 + 274360*x^9 + 4546826*x^8 - 20283562*x^7 + 50381510*x^6 - 78549692*x^5 + 119367930*x^4 - 132257608*x^3 + 19413309680*x^2 - 19356775495*x + 162278712581, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} + 6928 x^{12} - 41022 x^{11} + 20192 x^{10} + 274360 x^{9} + 4546826 x^{8} - 20283562 x^{7} + 50381510 x^{6} - 78549692 x^{5} + 119367930 x^{4} - 132257608 x^{3} + 19413309680 x^{2} - 19356775495 x + 162278712581 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7401566724659785057512888367473350870971393=43^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $477.90$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{81561764596051506638099318490841955} a^{14} - \frac{7}{81561764596051506638099318490841955} a^{13} - \frac{3756429537898590911924398679698252}{81561764596051506638099318490841955} a^{12} + \frac{22538577227391545471546392078189603}{81561764596051506638099318490841955} a^{11} - \frac{20353995217720673040346666095112332}{81561764596051506638099318490841955} a^{10} - \frac{23271883899767628316009278417001246}{81561764596051506638099318490841955} a^{9} - \frac{11356923006292547486366515691493033}{81561764596051506638099318490841955} a^{8} + \frac{25174521944825426838035453673855752}{81561764596051506638099318490841955} a^{7} - \frac{1336987447451101552739093272282835}{16312352919210301327619863698168391} a^{6} + \frac{18555075383619475425639417015718588}{81561764596051506638099318490841955} a^{5} + \frac{1949186885665858123406377661171464}{81561764596051506638099318490841955} a^{4} - \frac{11664437882990684968861897861820973}{81561764596051506638099318490841955} a^{3} + \frac{26999650018039037772617277092359058}{81561764596051506638099318490841955} a^{2} - \frac{18128404677615711144040694414754448}{81561764596051506638099318490841955} a - \frac{25249591605016306659356356029687562}{81561764596051506638099318490841955}$, $\frac{1}{841205447434882133198017617403321960648465} a^{15} + \frac{5156854}{841205447434882133198017617403321960648465} a^{14} - \frac{10390803892304429655293374734887000187074}{120172206776411733314002516771903137235495} a^{13} + \frac{4686780931565492619840988467672935170636}{120172206776411733314002516771903137235495} a^{12} + \frac{281979021049287722418090940401514937584747}{841205447434882133198017617403321960648465} a^{11} + \frac{91851924010635761127889375711966656218918}{841205447434882133198017617403321960648465} a^{10} + \frac{272633852318045613747678151516506111165582}{841205447434882133198017617403321960648465} a^{9} - \frac{127306760104064795580691957420159928182609}{841205447434882133198017617403321960648465} a^{8} - \frac{53717161767152790455704318675653844865118}{168241089486976426639603523480664392129693} a^{7} - \frac{17486159964177113187700544582479910334033}{120172206776411733314002516771903137235495} a^{6} + \frac{1788746955113114613752848445719951794694}{5427131918934723439987210434860141681603} a^{5} + \frac{6239970750810725684369882546841693018263}{27135659594673617199936052174300708408015} a^{4} - \frac{54184966035624652152786983397501006668622}{168241089486976426639603523480664392129693} a^{3} - \frac{42306882073444664406066323932740199343638}{841205447434882133198017617403321960648465} a^{2} + \frac{244170213379743153288357780217282027341719}{841205447434882133198017617403321960648465} a - \frac{375145807043416042352772811299499913788313}{841205447434882133198017617403321960648465}$

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

Class group and class number

$C_{8}\times C_{32}$, which has order $256$ (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:  $7$
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:  \( 7821277183660 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-4171}) \), 4.0.1687532377.1, 8.0.276233255772057202513.5

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed