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Properties

Label 16.8.52039206599...3569.3
Degree $16$
Signature $[8, 4]$
Discriminant $29^{14}\cdot 53^{10}$
Root discriminant $227.65$
Ramified primes $29, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31776832, 90002488, 35679363, -105159148, -89105387, 138673151, -34531314, -8854192, 1683238, 173397, 25904, -3848, -88, 143, -67, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 67*x^14 + 143*x^13 - 88*x^12 - 3848*x^11 + 25904*x^10 + 173397*x^9 + 1683238*x^8 - 8854192*x^7 - 34531314*x^6 + 138673151*x^5 - 89105387*x^4 - 105159148*x^3 + 35679363*x^2 + 90002488*x + 31776832)
 
gp: K = bnfinit(x^16 - 2*x^15 - 67*x^14 + 143*x^13 - 88*x^12 - 3848*x^11 + 25904*x^10 + 173397*x^9 + 1683238*x^8 - 8854192*x^7 - 34531314*x^6 + 138673151*x^5 - 89105387*x^4 - 105159148*x^3 + 35679363*x^2 + 90002488*x + 31776832, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 67 x^{14} + 143 x^{13} - 88 x^{12} - 3848 x^{11} + 25904 x^{10} + 173397 x^{9} + 1683238 x^{8} - 8854192 x^{7} - 34531314 x^{6} + 138673151 x^{5} - 89105387 x^{4} - 105159148 x^{3} + 35679363 x^{2} + 90002488 x + 31776832 \)

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:  \(52039206599103315275932373479504463569=29^{14}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.65$
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:  $29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{58} a^{8} - \frac{1}{58} a^{7} - \frac{5}{58} a^{6} - \frac{3}{29} a^{5} + \frac{5}{29} a^{4} - \frac{1}{58} a^{3} - \frac{9}{58} a^{2} - \frac{9}{58} a + \frac{8}{29}$, $\frac{1}{696} a^{9} - \frac{1}{696} a^{8} - \frac{25}{116} a^{7} - \frac{37}{87} a^{6} + \frac{329}{696} a^{5} + \frac{53}{116} a^{4} - \frac{45}{116} a^{3} - \frac{125}{696} a^{2} - \frac{245}{696} a + \frac{1}{3}$, $\frac{1}{696} a^{10} + \frac{5}{696} a^{8} + \frac{47}{348} a^{7} - \frac{17}{232} a^{6} - \frac{289}{696} a^{5} + \frac{9}{29} a^{4} + \frac{5}{24} a^{3} + \frac{157}{348} a^{2} - \frac{25}{696} a - \frac{7}{87}$, $\frac{1}{696} a^{11} + \frac{1}{232} a^{8} + \frac{33}{232} a^{7} + \frac{93}{232} a^{6} - \frac{157}{696} a^{5} - \frac{317}{696} a^{4} - \frac{41}{87} a^{3} + \frac{3}{29} a^{2} - \frac{55}{696} a + \frac{11}{87}$, $\frac{1}{221328} a^{12} - \frac{31}{55332} a^{11} - \frac{35}{55332} a^{10} + \frac{5}{7632} a^{9} + \frac{1657}{221328} a^{8} - \frac{1553}{221328} a^{7} - \frac{23195}{73776} a^{6} - \frac{997}{2544} a^{5} - \frac{10481}{55332} a^{4} - \frac{748}{13833} a^{3} - \frac{19081}{221328} a^{2} - \frac{31103}{110664} a + \frac{2116}{13833}$, $\frac{1}{8631792} a^{13} - \frac{7}{4315896} a^{12} + \frac{53}{81432} a^{11} - \frac{1693}{2877264} a^{10} + \frac{75}{319696} a^{9} + \frac{7663}{2877264} a^{8} - \frac{55357}{663984} a^{7} - \frac{1228333}{2877264} a^{6} + \frac{2118851}{4315896} a^{5} + \frac{454955}{4315896} a^{4} - \frac{1315447}{2877264} a^{3} + \frac{236893}{479544} a^{2} - \frac{9397}{20358} a + \frac{72170}{539487}$, $\frac{1}{3003863616} a^{14} + \frac{55}{1001287872} a^{13} - \frac{2387}{3003863616} a^{12} - \frac{698765}{3003863616} a^{11} - \frac{124159}{500643936} a^{10} + \frac{466267}{1001287872} a^{9} - \frac{4315027}{3003863616} a^{8} - \frac{278861771}{3003863616} a^{7} - \frac{685456951}{1501931808} a^{6} - \frac{449996123}{1001287872} a^{5} + \frac{289904369}{3003863616} a^{4} - \frac{447798053}{1001287872} a^{3} - \frac{259261039}{3003863616} a^{2} - \frac{16416221}{62580492} a - \frac{736634}{46935369}$, $\frac{1}{8689714911536849472919164487247808} a^{15} + \frac{1177402519194158877167045}{8689714911536849472919164487247808} a^{14} + \frac{334068295734608528665439149}{8689714911536849472919164487247808} a^{13} + \frac{3506322012018061186020499265}{2896571637178949824306388162415936} a^{12} + \frac{2257952195578646145439151735575}{4344857455768424736459582243623904} a^{11} - \frac{327274530875508319457132653319}{965523879059649941435462720805312} a^{10} + \frac{2120793758120511896767302483749}{8689714911536849472919164487247808} a^{9} - \frac{72540214592616813984576399851227}{8689714911536849472919164487247808} a^{8} + \frac{202508748916427297065402025802359}{1448285818589474912153194081207968} a^{7} - \frac{2763074456739574305118598041940473}{8689714911536849472919164487247808} a^{6} + \frac{3461851839554768029868250429923345}{8689714911536849472919164487247808} a^{5} + \frac{2224694364901664004221322482014777}{8689714911536849472919164487247808} a^{4} - \frac{3261009192754602991254089670795631}{8689714911536849472919164487247808} a^{3} + \frac{68042522126082683381352484709003}{1086214363942106184114895560905976} a^{2} + \frac{259097129789033397175880008940089}{543107181971053092057447780452988} a - \frac{22021110272121646678420227378298}{135776795492763273014361945113247}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 27920120795000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{1537}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{53}) \), 4.4.68508701.1 x2, 4.4.1292617.1 x2, \(\Q(\sqrt{29}, \sqrt{53})\), 8.8.4693442112707401.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$53$53.8.6.2$x^{8} + 477 x^{4} + 70225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$