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Properties

Label 16.0.28263286542...8329.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 79^{12}$
Root discriminant $69.29$
Ramified primes $3, 79$
Class number $405$ (GRH)
Class group $[9, 45]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27952369, -30305084, 37624698, -3024586, 4489289, 1254015, 1088971, 178093, -351, -2153, 2191, 1575, -19, -118, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 118*x^13 - 19*x^12 + 1575*x^11 + 2191*x^10 - 2153*x^9 - 351*x^8 + 178093*x^7 + 1088971*x^6 + 1254015*x^5 + 4489289*x^4 - 3024586*x^3 + 37624698*x^2 - 30305084*x + 27952369)
 
gp: K = bnfinit(x^16 - 2*x^15 - 118*x^13 - 19*x^12 + 1575*x^11 + 2191*x^10 - 2153*x^9 - 351*x^8 + 178093*x^7 + 1088971*x^6 + 1254015*x^5 + 4489289*x^4 - 3024586*x^3 + 37624698*x^2 - 30305084*x + 27952369, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 118 x^{13} - 19 x^{12} + 1575 x^{11} + 2191 x^{10} - 2153 x^{9} - 351 x^{8} + 178093 x^{7} + 1088971 x^{6} + 1254015 x^{5} + 4489289 x^{4} - 3024586 x^{3} + 37624698 x^{2} - 30305084 x + 27952369 \)

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:  \(282632865427655493724677478329=3^{14}\cdot 79^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.29$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{10} + \frac{1}{3} a^{3} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{2}{27} a^{2} - \frac{4}{27} a + \frac{1}{27}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{9} - \frac{1}{9} a^{6} - \frac{5}{27} a^{3} - \frac{2}{27}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{1}{9} a^{7} + \frac{4}{27} a^{4} + \frac{1}{3} a^{3} + \frac{7}{27} a + \frac{1}{3}$, $\frac{1}{100565406549} a^{14} - \frac{1629271729}{100565406549} a^{13} - \frac{466101233}{100565406549} a^{12} - \frac{740891528}{100565406549} a^{11} + \frac{1373292350}{100565406549} a^{10} - \frac{1478932082}{100565406549} a^{9} + \frac{910035359}{33521802183} a^{8} + \frac{2642880763}{33521802183} a^{7} + \frac{4746759677}{33521802183} a^{6} + \frac{8683068640}{100565406549} a^{5} + \frac{14661365900}{100565406549} a^{4} + \frac{43151587060}{100565406549} a^{3} + \frac{47560487767}{100565406549} a^{2} - \frac{4094079856}{100565406549} a - \frac{26564420702}{100565406549}$, $\frac{1}{60757450349264980935983168845406588504811} a^{15} + \frac{169214321389528997346409998886}{60757450349264980935983168845406588504811} a^{14} - \frac{197485895524862049513758181179407834}{11491857452102322855302282739815885853} a^{13} - \frac{170603293701447699835527158473049847359}{20252483449754993645327722948468862834937} a^{12} + \frac{121363906321549239959510595889117000249}{60757450349264980935983168845406588504811} a^{11} + \frac{2105628811891902905173453521710363292287}{60757450349264980935983168845406588504811} a^{10} - \frac{387865584428788515814424806532802215389}{60757450349264980935983168845406588504811} a^{9} - \frac{1045717573170153060018107508824083929514}{20252483449754993645327722948468862834937} a^{8} + \frac{847202339735863528682799575369049086323}{20252483449754993645327722948468862834937} a^{7} + \frac{7363492778322540258966770732735369611903}{60757450349264980935983168845406588504811} a^{6} - \frac{9064674768224595385213539598652439796268}{60757450349264980935983168845406588504811} a^{5} + \frac{5693699342448505966972462204288081429886}{60757450349264980935983168845406588504811} a^{4} + \frac{1372774783814323061106022808934706334915}{20252483449754993645327722948468862834937} a^{3} + \frac{29227365850327193732503501959448270655224}{60757450349264980935983168845406588504811} a^{2} - \frac{25533484345961333047235975239811069077456}{60757450349264980935983168845406588504811} a + \frac{2302583869994434165683662631599881469}{11491857452102322855302282739815885853}$

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

Class group and class number

$C_{9}\times C_{45}$, which has order $405$ (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:  \( \frac{447819570218099108578978}{16312280889859006552799587577253} a^{15} - \frac{776447297270410559508584}{16312280889859006552799587577253} a^{14} - \frac{85366470772241708639}{3085356703207680452581726419} a^{13} - \frac{16891266432459467908323241}{5437426963286335517599862525751} a^{12} - \frac{23510237484420382565948105}{16312280889859006552799587577253} a^{11} + \frac{715841357751691776996595633}{16312280889859006552799587577253} a^{10} + \frac{965633997582310631241198347}{16312280889859006552799587577253} a^{9} - \frac{463437248974121129477313745}{5437426963286335517599862525751} a^{8} + \frac{1044425641146379164165065177}{5437426963286335517599862525751} a^{7} + \frac{88458450765297933664826477899}{16312280889859006552799587577253} a^{6} + \frac{512818893870282039269007375079}{16312280889859006552799587577253} a^{5} + \frac{585039662020316113188539884696}{16312280889859006552799587577253} a^{4} + \frac{80830511397661107960970714730}{604158551476259501955540280639} a^{3} + \frac{1172355580190632694211379995574}{16312280889859006552799587577253} a^{2} + \frac{17490594491240727183611258109337}{16312280889859006552799587577253} a + \frac{434026318966723503117149939}{3085356703207680452581726419} \) (order $6$)
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:  \( 609706.825434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-79}) \), \(\Q(\sqrt{237}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-79})\), 4.2.168507.2 x2, 4.0.2133.1 x2, 8.0.28394609049.1, 8.2.531632265224427.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$79$79.8.6.2$x^{8} + 395 x^{4} + 56169$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
79.8.6.2$x^{8} + 395 x^{4} + 56169$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$