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Properties

Label 16.0.20974934542...4789.1
Degree $16$
Signature $[0, 8]$
Discriminant $19^{7}\cdot 31^{15}$
Root discriminant $90.70$
Ramified primes $19, 31$
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![1532768, -4451244, 7860421, -9260696, 7004864, -3845538, 1780247, -608662, 110277, -7146, 71, 158, 145, -66, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 - 66*x^13 + 145*x^12 + 158*x^11 + 71*x^10 - 7146*x^9 + 110277*x^8 - 608662*x^7 + 1780247*x^6 - 3845538*x^5 + 7004864*x^4 - 9260696*x^3 + 7860421*x^2 - 4451244*x + 1532768)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 - 66*x^13 + 145*x^12 + 158*x^11 + 71*x^10 - 7146*x^9 + 110277*x^8 - 608662*x^7 + 1780247*x^6 - 3845538*x^5 + 7004864*x^4 - 9260696*x^3 + 7860421*x^2 - 4451244*x + 1532768, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} - 66 x^{13} + 145 x^{12} + 158 x^{11} + 71 x^{10} - 7146 x^{9} + 110277 x^{8} - 608662 x^{7} + 1780247 x^{6} - 3845538 x^{5} + 7004864 x^{4} - 9260696 x^{3} + 7860421 x^{2} - 4451244 x + 1532768 \)

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:  \(20974934542740813230114132554789=19^{7}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.70$
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:  $19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{4} a^{6} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{19}{48} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{24} a^{9} - \frac{1}{48} a^{8} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{5}{24} a^{3} + \frac{7}{16} a^{2} + \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{192} a^{12} - \frac{1}{96} a^{11} + \frac{1}{192} a^{10} + \frac{1}{48} a^{9} - \frac{1}{192} a^{8} - \frac{3}{64} a^{6} + \frac{1}{32} a^{5} - \frac{25}{192} a^{4} - \frac{1}{3} a^{3} + \frac{89}{192} a^{2} + \frac{5}{48} a - \frac{1}{6}$, $\frac{1}{576} a^{13} + \frac{5}{576} a^{11} + \frac{1}{288} a^{10} - \frac{5}{576} a^{9} - \frac{11}{288} a^{8} - \frac{1}{64} a^{7} + \frac{11}{48} a^{6} + \frac{107}{576} a^{5} + \frac{11}{96} a^{4} - \frac{155}{576} a^{3} - \frac{43}{288} a^{2} - \frac{35}{72} a + \frac{4}{9}$, $\frac{1}{65664} a^{14} + \frac{13}{65664} a^{13} + \frac{7}{8208} a^{12} + \frac{19}{3456} a^{11} + \frac{13}{1368} a^{10} - \frac{89}{21888} a^{9} - \frac{935}{32832} a^{8} - \frac{1819}{21888} a^{7} - \frac{1351}{16416} a^{6} - \frac{3673}{65664} a^{5} - \frac{1277}{16416} a^{4} + \frac{10619}{65664} a^{3} - \frac{3421}{21888} a^{2} - \frac{25}{96} a + \frac{41}{108}$, $\frac{1}{7998307200702797042344235904} a^{15} + \frac{8864242353943133462399}{1999576800175699260586058976} a^{14} + \frac{15885943211464593726313}{296233600026029520086823552} a^{13} + \frac{370820210916322867365259}{296233600026029520086823552} a^{12} + \frac{71171994228098581182872167}{7998307200702797042344235904} a^{11} + \frac{1088202772605106005348467}{205084800018020436983185536} a^{10} + \frac{209369174656888758672051053}{7998307200702797042344235904} a^{9} + \frac{212668424201687696809868597}{7998307200702797042344235904} a^{8} + \frac{2733021851006271221251531}{22279407244297484797616256} a^{7} - \frac{36167272249216329274119961}{275803696575958518701525376} a^{6} - \frac{220909176126179531256563107}{888700800078088560260470656} a^{5} - \frac{573646966667609056367386939}{2666102400234265680781411968} a^{4} + \frac{1415874403094365305749911231}{3999153600351398521172117952} a^{3} + \frac{1035504332720029358236659329}{2666102400234265680781411968} a^{2} + \frac{23655588989585928759989617}{105240884219773645294003104} a - \frac{4946154892752039299905129}{13155110527471705661750388}$

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:  $7$
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:  \( 22232425610.9 \) (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

\(\Q(\sqrt{-31}) \), 4.0.566029.1, 8.0.188709020187349.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ $16$ R $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed