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Properties

Label 16.0.14046860872...8896.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 151^{8}\cdot 229^{8}$
Root discriminant $884.55$
Ramified primes $2, 151, 229$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1049

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7171989230975632144, 0, 266656622707871824, 0, 4535677481540424, 0, 43587599577536, 0, 252657800140, 0, 874215952, 0, 1805514, 0, 2066, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2066*x^14 + 1805514*x^12 + 874215952*x^10 + 252657800140*x^8 + 43587599577536*x^6 + 4535677481540424*x^4 + 266656622707871824*x^2 + 7171989230975632144)
 
gp: K = bnfinit(x^16 + 2066*x^14 + 1805514*x^12 + 874215952*x^10 + 252657800140*x^8 + 43587599577536*x^6 + 4535677481540424*x^4 + 266656622707871824*x^2 + 7171989230975632144, 1)
 

Normalized defining polynomial

\( x^{16} + 2066 x^{14} + 1805514 x^{12} + 874215952 x^{10} + 252657800140 x^{8} + 43587599577536 x^{6} + 4535677481540424 x^{4} + 266656622707871824 x^{2} + 7171989230975632144 \)

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:  \(140468608726897677774715983455580718320263888896=2^{36}\cdot 151^{8}\cdot 229^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $884.55$
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:  $2, 151, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{916} a^{10} + \frac{5}{916} a^{8} + \frac{39}{458} a^{6} + \frac{52}{229} a^{4} - \frac{62}{229} a^{2}$, $\frac{1}{916} a^{11} + \frac{5}{916} a^{9} + \frac{39}{458} a^{7} + \frac{52}{229} a^{5} - \frac{62}{229} a^{3}$, $\frac{1}{419528} a^{12} - \frac{28}{52441} a^{10} + \frac{5535}{209764} a^{8} - \frac{6157}{52441} a^{6} - \frac{14489}{104882} a^{4} - \frac{111}{229} a^{2}$, $\frac{1}{419528} a^{13} - \frac{28}{52441} a^{11} + \frac{5535}{209764} a^{9} - \frac{6157}{52441} a^{7} - \frac{14489}{104882} a^{5} - \frac{111}{229} a^{3}$, $\frac{1}{5030133307146171234817821718968829032459826940264} a^{14} - \frac{4554478517102981320246605078418758190903731}{5030133307146171234817821718968829032459826940264} a^{12} - \frac{165376103455076396310599317031637089579467703}{2515066653573085617408910859484414516229913470132} a^{10} - \frac{289789335340791075406721336798179637954828616367}{2515066653573085617408910859484414516229913470132} a^{8} - \frac{300911965254637241378346308864962011662178982463}{1257533326786542808704455429742207258114956735066} a^{6} - \frac{654366848052556418264510017998561632952674501}{2745705953682407879267369933934950345229163177} a^{4} - \frac{2369072209531428577195766842774030554788845}{11989982330490864101604235519366595394013813} a^{2} - \frac{10806409719855848313699469413985124455343}{52358001443191546295215002267976399100497}$, $\frac{1}{64219711932335168154919129886075040257414610546350488} a^{15} - \frac{7258075425158871242826025199366236551774586251}{16054927983083792038729782471518760064353652636587622} a^{13} + \frac{830589282795844772193036939240961593650184237695}{16054927983083792038729782471518760064353652636587622} a^{11} - \frac{1179963983682057024398828555763133925841521158452901}{32109855966167584077459564943037520128707305273175244} a^{9} - \frac{1636238033452806641016496850220322184011783969114410}{8027463991541896019364891235759380032176826318293811} a^{7} - \frac{7124457602417965533614338635022722095165028743516}{35054427910663301394606511946547511057540726280759} a^{5} + \frac{2110279246041428368509064387778581764463615822}{9004476730198638940304780875044313140904373563} a^{3} + \frac{244972282342973389266997296142447586266770120}{668454604425226471551009933955254687316045199} a$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 21186211828700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1049:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1049
Character table for t16n1049 is not computed

Intermediate fields

\(\Q(\sqrt{151}) \), 4.0.229.1, 8.0.6979410125322496.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
151Data not computed
229Data not computed