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Properties

Label 16.6.55013835857...3056.3
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}$
Root discriminant $11{,}124.47$
Ramified primes $2, 3, 7, 294337$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27830083703378571, 0, 8207228617425469476, 0, 1042122049007108184, 0, -2834713503199836, 0, -39818130261039, 0, -63913090068, 0, -13035650, 0, 4064, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4064*x^14 - 13035650*x^12 - 63913090068*x^10 - 39818130261039*x^8 - 2834713503199836*x^6 + 1042122049007108184*x^4 + 8207228617425469476*x^2 - 27830083703378571)
 
gp: K = bnfinit(x^16 + 4064*x^14 - 13035650*x^12 - 63913090068*x^10 - 39818130261039*x^8 - 2834713503199836*x^6 + 1042122049007108184*x^4 + 8207228617425469476*x^2 - 27830083703378571, 1)
 

Normalized defining polynomial

\( x^{16} + 4064 x^{14} - 13035650 x^{12} - 63913090068 x^{10} - 39818130261039 x^{8} - 2834713503199836 x^{6} + 1042122049007108184 x^{4} + 8207228617425469476 x^{2} - 27830083703378571 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55013835857676939099088270698740000784693073140191887871554093056=-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}124.47$
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:  $2, 3, 7, 294337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{45} a^{12} + \frac{1}{45} a^{10} - \frac{2}{15} a^{8} - \frac{1}{15} a^{6} - \frac{4}{15} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{45135} a^{13} - \frac{553}{15045} a^{11} - \frac{1606}{45135} a^{9} - \frac{551}{15045} a^{7} - \frac{973}{5015} a^{5} - \frac{2119}{5015} a^{3} - \frac{458}{5015} a$, $\frac{1}{12874764822541965730441323270221000244918414271318817111659463382328155} a^{14} + \frac{12792228965436829378939480461312229244188179159432042182213668341404}{12874764822541965730441323270221000244918414271318817111659463382328155} a^{12} + \frac{66161931359407631650272968640316675535067446118634632134482535108482}{12874764822541965730441323270221000244918414271318817111659463382328155} a^{10} - \frac{416005073950757660887235819862711374918533895225829503859915981801937}{4291588274180655243480441090073666748306138090439605703886487794109385} a^{8} - \frac{164590479493971541963203286428072471866851633558399463536657822919177}{4291588274180655243480441090073666748306138090439605703886487794109385} a^{6} - \frac{39582281921735665504817339627483602995250316132039040464577051373673}{1430529424726885081160147030024555582768712696813201901295495931369795} a^{4} - \frac{70227666210202937877992354694496557899205497444351025937808352088927}{286105884945377016232029406004911116553742539362640380259099186273959} a^{2} - \frac{927355345458450082294116563945113540288703410352255207866040544}{8057913405134230535287623175809045083780932326259649871264713945}$, $\frac{1}{759611124529975978096038072943039014450186442007810209587908339557361145} a^{15} - \frac{4037528972526524517062249303682542317796676097193862538909813204123}{759611124529975978096038072943039014450186442007810209587908339557361145} a^{13} + \frac{1591863680852844121775016845684572907621192912295236789268279075992129}{151922224905995195619207614588607802890037288401562041917581667911472229} a^{11} - \frac{111995468727383523271159087305511101171694804234851912211243885780283}{4964778591699189399320510280673457610785532300704641892731427055930465} a^{9} + \frac{84339324805059505195248460422079196246205151576040093525607889858318}{992955718339837879864102056134691522157106460140928378546285411186093} a^{7} + \frac{1076791661296500142929964068117169577283078415890812639369947224479618}{84401236058886219788448674771448779383354049111978912176434259950817905} a^{5} - \frac{9988979710191362020500285685559488637326021264182856566652422192183097}{84401236058886219788448674771448779383354049111978912176434259950817905} a^{3} + \frac{2092185105910321056632242241204579263732183938573536067789126478764}{8082087145349633226893486045336472219032275123238428820878508086835} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $10$
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:  \( 152492863296000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.2076841872.1, 8.6.975018691690245188654137344.1

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
Arithmetically equvalently 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
294337Data not computed