↑

Properties

Label 16.0.10995684793...9689.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 89^{9}$
Root discriminant $75.44$
Ramified primes $11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1545977, -5886424, 5869757, -1121147, 1091846, 333125, -392006, 123309, -50991, 5749, 6443, -2262, 753, -246, 32, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 32*x^14 - 246*x^13 + 753*x^12 - 2262*x^11 + 6443*x^10 + 5749*x^9 - 50991*x^8 + 123309*x^7 - 392006*x^6 + 333125*x^5 + 1091846*x^4 - 1121147*x^3 + 5869757*x^2 - 5886424*x + 1545977)
 
gp: K = bnfinit(x^16 - 5*x^15 + 32*x^14 - 246*x^13 + 753*x^12 - 2262*x^11 + 6443*x^10 + 5749*x^9 - 50991*x^8 + 123309*x^7 - 392006*x^6 + 333125*x^5 + 1091846*x^4 - 1121147*x^3 + 5869757*x^2 - 5886424*x + 1545977, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 32 x^{14} - 246 x^{13} + 753 x^{12} - 2262 x^{11} + 6443 x^{10} + 5749 x^{9} - 50991 x^{8} + 123309 x^{7} - 392006 x^{6} + 333125 x^{5} + 1091846 x^{4} - 1121147 x^{3} + 5869757 x^{2} - 5886424 x + 1545977 \)

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:  \(1099568479361490150598987419689=11^{12}\cdot 89^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.44$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{44} a^{12} - \frac{2}{11} a^{11} + \frac{5}{22} a^{10} - \frac{2}{11} a^{9} - \frac{5}{22} a^{8} - \frac{2}{11} a^{7} - \frac{1}{44} a^{6} + \frac{5}{22} a^{5} - \frac{9}{44} a^{4} + \frac{7}{22} a^{3} - \frac{17}{44} a^{2} - \frac{3}{22} a - \frac{19}{44}$, $\frac{1}{44} a^{13} - \frac{5}{22} a^{11} + \frac{3}{22} a^{10} - \frac{2}{11} a^{9} + \frac{1}{44} a^{7} + \frac{1}{22} a^{6} - \frac{17}{44} a^{5} - \frac{7}{22} a^{4} - \frac{15}{44} a^{3} - \frac{5}{22} a^{2} + \frac{21}{44} a - \frac{5}{11}$, $\frac{1}{33044} a^{14} - \frac{67}{16522} a^{13} - \frac{20}{8261} a^{12} + \frac{645}{16522} a^{11} + \frac{3655}{16522} a^{10} - \frac{515}{16522} a^{9} + \frac{5101}{33044} a^{8} + \frac{1669}{8261} a^{7} + \frac{995}{33044} a^{6} - \frac{2986}{8261} a^{5} + \frac{4537}{33044} a^{4} + \frac{851}{16522} a^{3} + \frac{1759}{33044} a^{2} + \frac{7395}{16522} a - \frac{4008}{8261}$, $\frac{1}{23612285708646215024531395654186497088036} a^{15} - \frac{8355643307376671751967526885035783}{23612285708646215024531395654186497088036} a^{14} + \frac{17681651851916141628765161510697547133}{5903071427161553756132848913546624272009} a^{13} + \frac{210161009075586312250486042261858876379}{23612285708646215024531395654186497088036} a^{12} + \frac{424157096876831801750210202218805863229}{11806142854323107512265697827093248544018} a^{11} + \frac{1277863159269397476406990488198040124232}{5903071427161553756132848913546624272009} a^{10} + \frac{2219474078000112418604706365124809159239}{23612285708646215024531395654186497088036} a^{9} - \frac{646835796014495648661002952846514353477}{23612285708646215024531395654186497088036} a^{8} + \frac{601409955212057256458793305425476304527}{23612285708646215024531395654186497088036} a^{7} + \frac{10808267658138501909471204084415129786}{5903071427161553756132848913546624272009} a^{6} + \frac{3338771792077785844015225876774658345897}{23612285708646215024531395654186497088036} a^{5} + \frac{131354015098959414339603758889376859596}{536642857014686705102986264867874933819} a^{4} + \frac{72775822304847107416133304234522994313}{2146571428058746820411945059471499735276} a^{3} - \frac{995225224064743091029977978642087097025}{5903071427161553756132848913546624272009} a^{2} - \frac{2640463134121603561109166102772858173827}{5903071427161553756132848913546624272009} a - \frac{5079120310026959825384313872848136773571}{23612285708646215024531395654186497088036}$

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:  $7$
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:  \( 124474223.474 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.10769.1, 8.0.10321451129.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$