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Properties

Label 16.2.79147628529...5423.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,3^{8}\cdot 47^{15}$
Root discriminant $64.00$
Ramified primes $3, 47$
Class number $1$ (GRH)
Class group Trivial (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![237988, 244766, -1038695, 1036385, -343558, -103855, 155388, -70843, 29369, -12980, 3573, -905, 392, -83, 22, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 22*x^14 - 83*x^13 + 392*x^12 - 905*x^11 + 3573*x^10 - 12980*x^9 + 29369*x^8 - 70843*x^7 + 155388*x^6 - 103855*x^5 - 343558*x^4 + 1036385*x^3 - 1038695*x^2 + 244766*x + 237988)
 
gp: K = bnfinit(x^16 - 5*x^15 + 22*x^14 - 83*x^13 + 392*x^12 - 905*x^11 + 3573*x^10 - 12980*x^9 + 29369*x^8 - 70843*x^7 + 155388*x^6 - 103855*x^5 - 343558*x^4 + 1036385*x^3 - 1038695*x^2 + 244766*x + 237988, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 22 x^{14} - 83 x^{13} + 392 x^{12} - 905 x^{11} + 3573 x^{10} - 12980 x^{9} + 29369 x^{8} - 70843 x^{7} + 155388 x^{6} - 103855 x^{5} - 343558 x^{4} + 1036385 x^{3} - 1038695 x^{2} + 244766 x + 237988 \)

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:  \(-79147628529732436014211365423=-\,3^{8}\cdot 47^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.00$
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:  $3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{20} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{3}{20} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{80} a^{11} + \frac{1}{20} a^{9} + \frac{7}{80} a^{8} - \frac{1}{8} a^{7} + \frac{1}{10} a^{6} - \frac{13}{80} a^{5} - \frac{1}{4} a^{4} - \frac{2}{5} a^{3} + \frac{29}{80} a^{2} - \frac{1}{8} a - \frac{1}{20}$, $\frac{1}{480} a^{12} - \frac{1}{160} a^{11} + \frac{1}{60} a^{10} + \frac{1}{160} a^{9} + \frac{7}{160} a^{8} - \frac{3}{16} a^{7} + \frac{67}{480} a^{6} - \frac{7}{160} a^{5} - \frac{1}{20} a^{4} - \frac{33}{160} a^{3} - \frac{13}{480} a^{2} - \frac{5}{16} a + \frac{11}{24}$, $\frac{1}{4800} a^{13} - \frac{1}{1200} a^{12} - \frac{13}{4800} a^{11} + \frac{23}{960} a^{10} + \frac{27}{800} a^{9} + \frac{187}{1600} a^{8} + \frac{1117}{4800} a^{7} + \frac{11}{120} a^{6} - \frac{137}{1600} a^{5} + \frac{19}{320} a^{4} - \frac{53}{2400} a^{3} - \frac{953}{4800} a^{2} - \frac{203}{480} a - \frac{31}{1200}$, $\frac{1}{57600} a^{14} + \frac{1}{11520} a^{13} - \frac{29}{57600} a^{12} - \frac{31}{28800} a^{11} - \frac{83}{57600} a^{10} + \frac{1333}{19200} a^{9} - \frac{67}{28800} a^{8} + \frac{9173}{57600} a^{7} + \frac{3449}{57600} a^{6} + \frac{187}{3200} a^{5} + \frac{1499}{57600} a^{4} - \frac{2447}{57600} a^{3} - \frac{21907}{57600} a^{2} - \frac{4217}{28800} a - \frac{7099}{14400}$, $\frac{1}{273351078619657534896190617600} a^{15} + \frac{20427723416539217209273}{3037234206885083721068784640} a^{14} + \frac{245112258582579223341571}{7593085517212709302671961600} a^{13} + \frac{8549010400387964106885953}{273351078619657534896190617600} a^{12} - \frac{1580749259852243108258530153}{273351078619657534896190617600} a^{11} - \frac{82639175376895661486053931}{17084442413728595931011913600} a^{10} - \frac{24899778137495534050949407259}{273351078619657534896190617600} a^{9} - \frac{2765035676898342039664493059}{91117026206552511632063539200} a^{8} - \frac{14005190879311673695021010023}{136675539309828767448095308800} a^{7} - \frac{50048648774418921667852711309}{273351078619657534896190617600} a^{6} + \frac{21431881526891024350629618809}{273351078619657534896190617600} a^{5} - \frac{2209355743913045653067038513}{11389628275819063954007942400} a^{4} + \frac{7142865933148361436363418403}{45558513103276255816031769600} a^{3} - \frac{3068790305631705167924967569}{273351078619657534896190617600} a^{2} - \frac{58944553325566720655968215763}{136675539309828767448095308800} a + \frac{2003637073955502744074847397}{13667553930982876744809530880}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2086348151.54 \) (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

\(\Q(\sqrt{141}) \), 4.2.934407.1, 8.2.41036472757503.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47Data not computed