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Properties

Label 16.0.78810998787...249.14
Degree $16$
Signature $[0, 8]$
Discriminant $71^{8}\cdot 73^{14}$
Root discriminant $359.78$
Ramified primes $71, 73$
Class number $89600$ (GRH)
Class group $[8, 40, 280]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![259398713344, 0, 24471001088, 0, 2825382976, 0, 74158208, 0, -3371040, 0, -108753, 0, 1392, 0, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 72*x^14 + 1392*x^12 - 108753*x^10 - 3371040*x^8 + 74158208*x^6 + 2825382976*x^4 + 24471001088*x^2 + 259398713344)
 
gp: K = bnfinit(x^16 + 72*x^14 + 1392*x^12 - 108753*x^10 - 3371040*x^8 + 74158208*x^6 + 2825382976*x^4 + 24471001088*x^2 + 259398713344, 1)
 

Normalized defining polynomial

\( x^{16} + 72 x^{14} + 1392 x^{12} - 108753 x^{10} - 3371040 x^{8} + 74158208 x^{6} + 2825382976 x^{4} + 24471001088 x^{2} + 259398713344 \)

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:  \(78810998787104838676665578559275236411249=71^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $359.78$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{148} a^{8} - \frac{2}{37} a^{6} + \frac{2}{37} a^{4} - \frac{17}{148} a^{2} + \frac{12}{37}$, $\frac{1}{296} a^{9} - \frac{1}{37} a^{7} + \frac{1}{37} a^{5} - \frac{17}{296} a^{3} + \frac{6}{37} a$, $\frac{1}{1184} a^{10} - \frac{1}{4} a^{7} + \frac{15}{74} a^{6} + \frac{47}{1184} a^{4} - \frac{11}{148} a^{2} + \frac{1}{4} a + \frac{12}{37}$, $\frac{1}{2368} a^{11} - \frac{1}{296} a^{8} + \frac{15}{148} a^{7} + \frac{1}{37} a^{6} + \frac{47}{2368} a^{5} - \frac{1}{37} a^{4} - \frac{11}{296} a^{3} + \frac{17}{296} a^{2} + \frac{6}{37} a - \frac{6}{37}$, $\frac{1}{4736} a^{12} - \frac{1}{592} a^{9} - \frac{1}{296} a^{8} + \frac{1}{74} a^{7} - \frac{273}{4736} a^{6} - \frac{1}{74} a^{5} + \frac{29}{592} a^{4} + \frac{17}{592} a^{3} + \frac{31}{74} a + \frac{15}{37}$, $\frac{1}{18944} a^{13} + \frac{1}{1184} a^{9} - \frac{1}{296} a^{8} - \frac{2897}{18944} a^{7} + \frac{1}{37} a^{6} - \frac{235}{2368} a^{5} - \frac{1}{37} a^{4} + \frac{205}{592} a^{3} + \frac{17}{296} a^{2} + \frac{17}{296} a + \frac{25}{74}$, $\frac{1}{676408041831480099408355328} a^{14} + \frac{7625586639885718015427}{84551005228935012426044416} a^{12} + \frac{13557505053333099187919}{42275502614467506213022208} a^{10} - \frac{1}{592} a^{9} - \frac{973477982048829560714705}{676408041831480099408355328} a^{8} + \frac{1}{74} a^{7} + \frac{263900496988971052871933}{1142581151742365032784384} a^{6} - \frac{1}{74} a^{5} - \frac{1308009804750386486785341}{5284437826808438276627776} a^{4} + \frac{17}{592} a^{3} - \frac{3680598379955578030626351}{10568875653616876553255552} a^{2} - \frac{3}{37} a + \frac{199881505641691251619183}{660554728351054784578472}$, $\frac{1}{21531420787579674524366766800896} a^{15} - \frac{1}{1352816083662960198816710656} a^{14} + \frac{42622331980530906550145499}{2691427598447459315545845850112} a^{13} + \frac{10227243856088735621829}{169102010457870024852088832} a^{12} - \frac{124599199356848353288858961}{1345713799223729657772922925056} a^{11} - \frac{13557505053333099187919}{84551005228935012426044416} a^{10} + \frac{30960690571932558828830522927}{21531420787579674524366766800896} a^{9} + \frac{3258640285533559626283473}{1352816083662960198816710656} a^{8} + \frac{4622312741248144168899418401}{36370643222262963723592511488} a^{7} - \frac{391524109588572214684209}{2285162303484730065568768} a^{6} - \frac{18240045289780187813565711037}{168214224902966207221615365632} a^{5} - \frac{789697778526611815592239}{10568875653616876553255552} a^{4} + \frac{82657066937641281696901124369}{336428449805932414443230731264} a^{3} + \frac{2466605906229315183292943}{21137751307233753106511104} a^{2} - \frac{8805637349311773667510353793}{21026778112870775902701920704} a - \frac{378409810601435787991743}{1321109456702109569156944}$

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

Class group and class number

$C_{8}\times C_{40}\times C_{280}$, which has order $89600$ (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:  \( 425039068999 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{-5183}) \), \(\Q(\sqrt{-71}) \), 4.0.1961034697.2, 4.4.389017.1, \(\Q(\sqrt{-71}, \sqrt{73})\), 8.0.280732967047165372057.3 x2, 8.4.55689935934767977.2 x2, 8.0.3845657082837881809.4

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$