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Properties

Label 16.0.54646496237...1129.7
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 89^{15}$
Root discriminant $406.07$
Ramified primes $11, 89$
Class number $128$ (GRH)
Class group $[4, 32]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1934079109825, 542456662935, 250721809619, 99957289357, 18102947155, 4874750016, 1300334847, 162727605, 31631282, 7217528, 586457, 73652, 16099, -440, -260, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 260*x^14 - 440*x^13 + 16099*x^12 + 73652*x^11 + 586457*x^10 + 7217528*x^9 + 31631282*x^8 + 162727605*x^7 + 1300334847*x^6 + 4874750016*x^5 + 18102947155*x^4 + 99957289357*x^3 + 250721809619*x^2 + 542456662935*x + 1934079109825)
 
gp: K = bnfinit(x^16 - 3*x^15 - 260*x^14 - 440*x^13 + 16099*x^12 + 73652*x^11 + 586457*x^10 + 7217528*x^9 + 31631282*x^8 + 162727605*x^7 + 1300334847*x^6 + 4874750016*x^5 + 18102947155*x^4 + 99957289357*x^3 + 250721809619*x^2 + 542456662935*x + 1934079109825, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 260 x^{14} - 440 x^{13} + 16099 x^{12} + 73652 x^{11} + 586457 x^{10} + 7217528 x^{9} + 31631282 x^{8} + 162727605 x^{7} + 1300334847 x^{6} + 4874750016 x^{5} + 18102947155 x^{4} + 99957289357 x^{3} + 250721809619 x^{2} + 542456662935 x + 1934079109825 \)

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:  \(546464962373097060033371075256437529131129=11^{12}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $406.07$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} - \frac{3}{11} a^{9} + \frac{1}{22} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{22} a^{5} + \frac{9}{22} a^{4} - \frac{2}{11} a^{3} + \frac{9}{22} a^{2} - \frac{7}{22} a + \frac{1}{22}$, $\frac{1}{22} a^{11} + \frac{9}{22} a^{9} - \frac{4}{11} a^{8} - \frac{1}{11} a^{7} + \frac{5}{22} a^{6} - \frac{9}{22} a^{5} + \frac{3}{11} a^{4} + \frac{7}{22} a^{3} + \frac{3}{22} a^{2} + \frac{3}{22} a + \frac{3}{11}$, $\frac{1}{110} a^{12} + \frac{1}{110} a^{11} - \frac{3}{10} a^{9} + \frac{3}{110} a^{8} + \frac{41}{110} a^{7} + \frac{3}{55} a^{6} + \frac{23}{55} a^{5} - \frac{1}{55} a^{4} + \frac{12}{55} a^{3} - \frac{53}{110} a^{2} + \frac{3}{55} a - \frac{5}{22}$, $\frac{1}{110} a^{13} - \frac{1}{110} a^{11} + \frac{1}{55} a^{10} + \frac{23}{55} a^{9} - \frac{37}{110} a^{8} + \frac{5}{22} a^{7} + \frac{5}{11} a^{6} - \frac{43}{110} a^{5} + \frac{1}{10} a^{4} + \frac{3}{110} a^{3} + \frac{2}{5} a^{2} + \frac{27}{55} a - \frac{5}{11}$, $\frac{1}{550} a^{14} + \frac{1}{275} a^{13} + \frac{1}{550} a^{12} - \frac{4}{275} a^{11} + \frac{1}{55} a^{10} - \frac{81}{550} a^{9} - \frac{113}{550} a^{8} - \frac{59}{275} a^{7} - \frac{71}{550} a^{6} + \frac{7}{550} a^{5} + \frac{261}{550} a^{4} + \frac{39}{275} a^{3} - \frac{122}{275} a^{2} - \frac{1}{55} a - \frac{5}{11}$, $\frac{1}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{15} + \frac{87381151579054008959183074349491010021238999539984674465882783372289069}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{14} + \frac{1402672964342360994499073913328567353315955858471784013410731946839852913}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{13} + \frac{750928931474214046008802897350164802942249166976100486425872001681761889}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{12} - \frac{1967979319131559975564367692277808905417349318555921948022166537126953134}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{11} + \frac{2346827811473480564748415849459686301916645579883133440065686072902263467}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{10} + \frac{5120033595872140027763648832894650631608811090066805681860054829541867151}{42410084751755823030269857387912605928433425039353199956941589500635050050} a^{9} - \frac{215064542375152722251513947794246469706944543151147930857092404009257614041}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{8} + \frac{113213850370422349534254592986754780395099326773586072718969390477030746031}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{7} - \frac{201209922171595302042623090265430649204627676757777898298521162863102874899}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{6} + \frac{85278514489479322358648496752488218389025118007964345253142102124717906709}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{5} - \frac{36020690139916538370319678698497078868484400958377508774653559733411402161}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{4} - \frac{107798447094322265691186837463479200699387087774498476217367196686232618958}{233255466134657026666484215633519332606383837716442599763178742253492775275} a^{3} + \frac{161200244714373229706051918160484966184868944080201253107048614983813689791}{466510932269314053332968431267038665212767675432885199526357484506985550550} a^{2} - \frac{4618885730389054609952984120084623779205465541570324541014624550748320741}{18660437290772562133318737250681546608510707017315407981054299380279422022} a - \frac{103301026458204600054182778876368980255193615192057412797356951047252543}{18660437290772562133318737250681546608510707017315407981054299380279422022}$

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

Class group and class number

$C_{4}\times C_{32}$, which has order $128$ (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:  \( 13163017928100 \) (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

\(\Q(\sqrt{-979}) \), 4.0.85301249.1, 8.0.647590974205440089.5

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ $16$

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}$
89Data not computed