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Properties

Label 16.0.48789140945...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}\cdot 41^{6}$
Root discriminant $110.41$
Ramified primes $5, 29, 41$
Class number $144$ (GRH)
Class group $[2, 2, 36]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1983836, 1779336, -516395, -1268322, 2587094, -2981908, 2259165, -1109520, 390916, -123415, 34055, -6856, 1331, -358, 67, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 67*x^14 - 358*x^13 + 1331*x^12 - 6856*x^11 + 34055*x^10 - 123415*x^9 + 390916*x^8 - 1109520*x^7 + 2259165*x^6 - 2981908*x^5 + 2587094*x^4 - 1268322*x^3 - 516395*x^2 + 1779336*x + 1983836)
 
gp: K = bnfinit(x^16 - 7*x^15 + 67*x^14 - 358*x^13 + 1331*x^12 - 6856*x^11 + 34055*x^10 - 123415*x^9 + 390916*x^8 - 1109520*x^7 + 2259165*x^6 - 2981908*x^5 + 2587094*x^4 - 1268322*x^3 - 516395*x^2 + 1779336*x + 1983836, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 67 x^{14} - 358 x^{13} + 1331 x^{12} - 6856 x^{11} + 34055 x^{10} - 123415 x^{9} + 390916 x^{8} - 1109520 x^{7} + 2259165 x^{6} - 2981908 x^{5} + 2587094 x^{4} - 1268322 x^{3} - 516395 x^{2} + 1779336 x + 1983836 \)

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:  \(487891409452798143616318603515625=5^{12}\cdot 29^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.41$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{53732} a^{14} - \frac{67}{1414} a^{13} + \frac{661}{26866} a^{12} - \frac{1137}{3838} a^{11} + \frac{13359}{53732} a^{10} + \frac{15753}{53732} a^{9} + \frac{3369}{7676} a^{8} + \frac{17937}{53732} a^{7} - \frac{289}{1919} a^{6} - \frac{169}{2828} a^{5} - \frac{7177}{26866} a^{4} + \frac{3195}{53732} a^{3} + \frac{23911}{53732} a^{2} - \frac{444}{1919} a + \frac{2829}{13433}$, $\frac{1}{75447936771615802270777479095788093163921541332} a^{15} - \frac{1727295568184312029850673047998860522106}{2694569170414850081099195681992431898711483619} a^{14} - \frac{1616434243791656790237505872852516014430079965}{75447936771615802270777479095788093163921541332} a^{13} - \frac{1662758727196681712292947240356049605065664009}{75447936771615802270777479095788093163921541332} a^{12} - \frac{9230716688028007828059037854689346494453310202}{18861984192903950567694369773947023290980385333} a^{11} + \frac{6067962333211120823644509261767745444824184157}{18861984192903950567694369773947023290980385333} a^{10} + \frac{25801054086443823532101081106239431599180412999}{75447936771615802270777479095788093163921541332} a^{9} - \frac{18375024662181686023634163070022661856113134749}{37723968385807901135388739547894046581960770666} a^{8} + \frac{14870107268228973181928249491117626221908854327}{37723968385807901135388739547894046581960770666} a^{7} - \frac{14974694071765178087812200706115136023280604121}{37723968385807901135388739547894046581960770666} a^{6} - \frac{5617309170904771917021290472525164093797105333}{75447936771615802270777479095788093163921541332} a^{5} - \frac{35245668071192396892621577549079554470979845115}{75447936771615802270777479095788093163921541332} a^{4} + \frac{12213065637367997993675603056993510127013465}{567277720087336859178778038314196189202417604} a^{3} - \frac{32290040791331927930964990736257577296214775627}{75447936771615802270777479095788093163921541332} a^{2} + \frac{4517475173638009674039001916823773228661520724}{18861984192903950567694369773947023290980385333} a - \frac{5578336039117404355214374699051232866049600533}{18861984192903950567694369773947023290980385333}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{36}$, which has order $144$ (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:  \( 168237223.318 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

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

\(\Q(\sqrt{5}) \), 4.0.3625.1, 4.0.862025.1, 4.4.148625.2, 8.0.18577177515625.8

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$