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Properties

Label 16.12.2370318527...9776.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{66}\cdot 17^{6}\cdot 191^{4}$
Root discriminant $187.68$
Ramified primes $2, 17, 191$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-61147198, 144814432, 1233840416, 2111300176, 1391830936, 286004864, -58032376, -35641232, -4977480, 751200, 281136, 6448, -780, -192, -112, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 112*x^14 - 192*x^13 - 780*x^12 + 6448*x^11 + 281136*x^10 + 751200*x^9 - 4977480*x^8 - 35641232*x^7 - 58032376*x^6 + 286004864*x^5 + 1391830936*x^4 + 2111300176*x^3 + 1233840416*x^2 + 144814432*x - 61147198)
 
gp: K = bnfinit(x^16 - 112*x^14 - 192*x^13 - 780*x^12 + 6448*x^11 + 281136*x^10 + 751200*x^9 - 4977480*x^8 - 35641232*x^7 - 58032376*x^6 + 286004864*x^5 + 1391830936*x^4 + 2111300176*x^3 + 1233840416*x^2 + 144814432*x - 61147198, 1)
 

Normalized defining polynomial

\( x^{16} - 112 x^{14} - 192 x^{13} - 780 x^{12} + 6448 x^{11} + 281136 x^{10} + 751200 x^{9} - 4977480 x^{8} - 35641232 x^{7} - 58032376 x^{6} + 286004864 x^{5} + 1391830936 x^{4} + 2111300176 x^{3} + 1233840416 x^{2} + 144814432 x - 61147198 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2370318527000656631345659186329419776=2^{66}\cdot 17^{6}\cdot 191^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187.68$
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:  $2, 17, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{5}{17} a^{11} - \frac{2}{17} a^{10} + \frac{2}{17} a^{9} - \frac{5}{17} a^{8} - \frac{3}{17} a^{7} - \frac{8}{17} a^{6} - \frac{2}{17} a^{5} + \frac{8}{17} a^{4} + \frac{5}{17} a^{3} + \frac{5}{17} a^{2}$, $\frac{1}{17} a^{13} + \frac{7}{17} a^{11} - \frac{5}{17} a^{10} + \frac{2}{17} a^{9} + \frac{5}{17} a^{8} + \frac{7}{17} a^{7} + \frac{4}{17} a^{6} + \frac{1}{17} a^{5} - \frac{1}{17} a^{4} - \frac{3}{17} a^{3} - \frac{8}{17} a^{2}$, $\frac{1}{17} a^{14} - \frac{6}{17} a^{11} - \frac{1}{17} a^{10} + \frac{8}{17} a^{9} + \frac{8}{17} a^{8} + \frac{8}{17} a^{7} + \frac{6}{17} a^{6} - \frac{4}{17} a^{5} - \frac{8}{17} a^{4} + \frac{8}{17} a^{3} - \frac{1}{17} a^{2}$, $\frac{1}{219708376540806369339366734378111002693372210466354050525185435071} a^{15} - \frac{12244213994463894744772169509218153471194307167023385600018678}{1846288878494171170919048188051352963809850508120622273320886009} a^{14} - \frac{58293433844450189199007272793843260375220025142529324543440624}{31386910934400909905623819196873000384767458638050578646455062153} a^{13} + \frac{2436309675996794304395109353933518505137983074771990442839674436}{219708376540806369339366734378111002693372210466354050525185435071} a^{12} - \frac{105227383833235479105207101906293628120937115535709440076742770966}{219708376540806369339366734378111002693372210466354050525185435071} a^{11} - \frac{18064111432072432296395753192029230792253054565556653171572238960}{219708376540806369339366734378111002693372210466354050525185435071} a^{10} + \frac{108262592452796519154263360052060210598759542571625418943285915329}{219708376540806369339366734378111002693372210466354050525185435071} a^{9} - \frac{25656782092269744842468646091603778500968555329698355880753093818}{219708376540806369339366734378111002693372210466354050525185435071} a^{8} - \frac{101518928011312944974993000948296291579812367729652511317612703005}{219708376540806369339366734378111002693372210466354050525185435071} a^{7} - \frac{42692575167047832647100500007549912738470838554800288375383292568}{219708376540806369339366734378111002693372210466354050525185435071} a^{6} - \frac{49001776446388435005283526152272946821478352756573381208242360948}{219708376540806369339366734378111002693372210466354050525185435071} a^{5} - \frac{89466851719199222140358199912897708433783185916944423522077619399}{219708376540806369339366734378111002693372210466354050525185435071} a^{4} + \frac{2783759916143476188707698141743830689395717998816044447127777207}{12924022149459198196433337316359470746668953556844355913246202063} a^{3} + \frac{805965987292813572582935708074768472111286432083005375633167038}{12924022149459198196433337316359470746668953556844355913246202063} a^{2} - \frac{26277226997095233540465205704973098112814905869206699053548660}{760236597027011658613725724491733573333467856284962112543894239} a - \frac{18865655200447683516713547841102962861192506170542819334453301}{108605228146715951230532246355961939047638265183566016077699177}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $13$
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:  \( 2288153594680 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.195584.1, 4.4.391168.1, 8.8.2448198467584.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
191Data not computed