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Properties

Label 16.2.25586390708...4027.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 19^{7}$
Root discriminant $51.64$
Ramified primes $17, 19$
Class number $4$ (GRH)
Class group $[4]$ (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![-288, -3120, -10408, -14516, -21710, 10311, -1866, -189, -2379, 858, -327, 393, -18, 9, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 13*x^14 + 9*x^13 - 18*x^12 + 393*x^11 - 327*x^10 + 858*x^9 - 2379*x^8 - 189*x^7 - 1866*x^6 + 10311*x^5 - 21710*x^4 - 14516*x^3 - 10408*x^2 - 3120*x - 288)
 
gp: K = bnfinit(x^16 - 2*x^15 - 13*x^14 + 9*x^13 - 18*x^12 + 393*x^11 - 327*x^10 + 858*x^9 - 2379*x^8 - 189*x^7 - 1866*x^6 + 10311*x^5 - 21710*x^4 - 14516*x^3 - 10408*x^2 - 3120*x - 288, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 13 x^{14} + 9 x^{13} - 18 x^{12} + 393 x^{11} - 327 x^{10} + 858 x^{9} - 2379 x^{8} - 189 x^{7} - 1866 x^{6} + 10311 x^{5} - 21710 x^{4} - 14516 x^{3} - 10408 x^{2} - 3120 x - 288 \)

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:  \(-2558639070806765618458574027=-\,17^{15}\cdot 19^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.64$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{72} a^{10} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{18} a^{6} - \frac{1}{12} a^{5} + \frac{1}{8} a^{4} - \frac{11}{24} a^{3} - \frac{7}{36} a^{2} + \frac{1}{3} a$, $\frac{1}{72} a^{11} - \frac{1}{24} a^{9} + \frac{1}{36} a^{7} - \frac{1}{12} a^{6} + \frac{1}{24} a^{5} - \frac{1}{4} a^{4} - \frac{35}{72} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a$, $\frac{1}{144} a^{12} + \frac{1}{48} a^{9} - \frac{1}{36} a^{8} - \frac{1}{16} a^{6} - \frac{1}{12} a^{5} - \frac{2}{9} a^{4} - \frac{3}{16} a^{3} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{288} a^{13} - \frac{1}{288} a^{12} - \frac{1}{288} a^{10} + \frac{5}{288} a^{9} + \frac{1}{72} a^{8} + \frac{5}{96} a^{7} + \frac{13}{288} a^{6} + \frac{1}{72} a^{5} - \frac{31}{288} a^{4} + \frac{5}{96} a^{3} - \frac{7}{18} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{864} a^{14} - \frac{1}{288} a^{12} + \frac{1}{288} a^{11} - \frac{7}{288} a^{9} + \frac{1}{288} a^{8} + \frac{5}{72} a^{7} - \frac{7}{288} a^{6} - \frac{5}{288} a^{5} + \frac{11}{72} a^{4} - \frac{89}{288} a^{3} + \frac{19}{54} a^{2} - \frac{11}{36} a - \frac{5}{12}$, $\frac{1}{2222248106885766336} a^{15} + \frac{3366455379527}{1111124053442883168} a^{14} - \frac{164054002940233}{246916456320640704} a^{13} + \frac{1400960799303527}{740749368961922112} a^{12} + \frac{227109079242703}{41152742720106784} a^{11} + \frac{104963919285023}{82305485440213568} a^{10} - \frac{29823897949611769}{740749368961922112} a^{9} - \frac{13834964072498395}{370374684480961056} a^{8} + \frac{5573322641029907}{740749368961922112} a^{7} - \frac{18179068297704211}{740749368961922112} a^{6} + \frac{13390242972037055}{370374684480961056} a^{5} - \frac{139187967991051207}{740749368961922112} a^{4} + \frac{58636078695194867}{1111124053442883168} a^{3} - \frac{132721849393123681}{277781013360720792} a^{2} - \frac{19522014770937407}{92593671120240264} a + \frac{466810045794029}{15432278520040044}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 200097097.264 \) (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

\(\Q(\sqrt{17}) \), 4.2.93347.1, 8.2.2814512958107.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R R $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
17Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$