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Properties

Label 16.2.69542802636...1743.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,7^{8}\cdot 47^{15}$
Root discriminant $97.76$
Ramified primes $7, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52875, -17625, -155335, -768591, 91368, 252484, 303620, -82203, -112706, 10951, 11186, -705, -470, -47, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 47*x^13 - 470*x^12 - 705*x^11 + 11186*x^10 + 10951*x^9 - 112706*x^8 - 82203*x^7 + 303620*x^6 + 252484*x^5 + 91368*x^4 - 768591*x^3 - 155335*x^2 - 17625*x + 52875)
 
gp: K = bnfinit(x^16 - 47*x^13 - 470*x^12 - 705*x^11 + 11186*x^10 + 10951*x^9 - 112706*x^8 - 82203*x^7 + 303620*x^6 + 252484*x^5 + 91368*x^4 - 768591*x^3 - 155335*x^2 - 17625*x + 52875, 1)
 

Normalized defining polynomial

\( x^{16} - 47 x^{13} - 470 x^{12} - 705 x^{11} + 11186 x^{10} + 10951 x^{9} - 112706 x^{8} - 82203 x^{7} + 303620 x^{6} + 252484 x^{5} + 91368 x^{4} - 768591 x^{3} - 155335 x^{2} - 17625 x + 52875 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-69542802636157609642914448041743=-\,7^{8}\cdot 47^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.76$
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:  $7, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{45} a^{12} + \frac{2}{45} a^{11} + \frac{1}{45} a^{10} - \frac{1}{45} a^{9} - \frac{1}{9} a^{8} + \frac{1}{15} a^{7} + \frac{7}{45} a^{6} + \frac{16}{45} a^{5} - \frac{4}{9} a^{4} - \frac{11}{45} a^{3} - \frac{2}{45} a^{2} + \frac{2}{5} a$, $\frac{1}{135} a^{13} + \frac{2}{135} a^{11} - \frac{2}{15} a^{10} + \frac{17}{135} a^{9} - \frac{17}{135} a^{8} + \frac{7}{45} a^{7} + \frac{52}{135} a^{6} - \frac{19}{45} a^{5} - \frac{61}{135} a^{4} - \frac{4}{27} a^{3} - \frac{58}{135} a^{2} - \frac{22}{45} a - \frac{1}{3}$, $\frac{1}{665145} a^{14} + \frac{1042}{665145} a^{13} + \frac{368}{51165} a^{12} - \frac{245}{133029} a^{11} + \frac{43958}{665145} a^{10} + \frac{7387}{44343} a^{9} - \frac{84398}{665145} a^{8} + \frac{21227}{133029} a^{7} - \frac{145439}{665145} a^{6} - \frac{139873}{665145} a^{5} + \frac{43736}{221715} a^{4} - \frac{1699}{4927} a^{3} - \frac{223396}{665145} a^{2} - \frac{84427}{221715} a - \frac{3421}{14781}$, $\frac{1}{13938362462475524304526233223186378425} a^{15} - \frac{832726458660543805511418710923}{2787672492495104860905246644637275685} a^{14} + \frac{219155233352685870795711109185581}{214436345576546527761942049587482745} a^{13} + \frac{126735722771679080715838196268295493}{13938362462475524304526233223186378425} a^{12} - \frac{2262815729680446765397603251783137}{929224164165034953635082214879091895} a^{11} - \frac{124766205206279313542804983771347514}{929224164165034953635082214879091895} a^{10} - \frac{2070408070755897789987794383785347224}{13938362462475524304526233223186378425} a^{9} + \frac{178313863686462384820794402033486659}{1548706940275058256058470358131819825} a^{8} - \frac{228805681099365453857141902277177936}{13938362462475524304526233223186378425} a^{7} + \frac{6265334153086635312400776964285014847}{13938362462475524304526233223186378425} a^{6} + \frac{18946566261911626013080345770316223}{185844832833006990727016442975818379} a^{5} - \frac{6544236246098544285947468754824334431}{13938362462475524304526233223186378425} a^{4} + \frac{3463225123443000602131550249313496843}{13938362462475524304526233223186378425} a^{3} - \frac{123812540963333952790415316182361344}{733598024340817068659275432799283075} a^{2} + \frac{112109197436867132472006856910671289}{929224164165034953635082214879091895} a - \frac{96305204042376154346061092830003}{4765252123923256172487601101944061}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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:  \( 4950271109.28 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{329}) \), 4.2.5087327.1, 8.2.1216402112231663.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.4.2.1$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$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47Data not computed