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Properties

Label 16.0.21067971336...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 149^{10}$
Root discriminant $51.02$
Ramified primes $5, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27869, 9424, 40421, -14428, 15301, 38884, 13090, -34526, 40781, -20362, 9994, -3072, 1092, -182, 56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869)
 
gp: K = bnfinit(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 56 x^{14} - 182 x^{13} + 1092 x^{12} - 3072 x^{11} + 9994 x^{10} - 20362 x^{9} + 40781 x^{8} - 34526 x^{7} + 13090 x^{6} + 38884 x^{5} + 15301 x^{4} - 14428 x^{3} + 40421 x^{2} + 9424 x + 27869 \)

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:  \(2106797133618518949609765625=5^{8}\cdot 149^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.02$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{4} - \frac{3}{10} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{580} a^{12} - \frac{23}{580} a^{11} - \frac{13}{580} a^{10} - \frac{13}{116} a^{9} + \frac{3}{290} a^{8} - \frac{233}{580} a^{7} - \frac{69}{145} a^{6} + \frac{177}{580} a^{5} + \frac{15}{58} a^{4} - \frac{121}{580} a^{3} - \frac{99}{580} a^{2} + \frac{14}{29} a - \frac{3}{20}$, $\frac{1}{580} a^{13} - \frac{1}{29} a^{11} - \frac{4}{145} a^{10} + \frac{19}{580} a^{9} + \frac{137}{580} a^{8} - \frac{9}{580} a^{7} + \frac{93}{580} a^{6} - \frac{49}{116} a^{5} - \frac{209}{580} a^{4} - \frac{68}{145} a^{3} + \frac{207}{580} a^{2} - \frac{259}{580} a - \frac{1}{4}$, $\frac{1}{259260} a^{14} + \frac{1}{12963} a^{13} + \frac{47}{129630} a^{12} + \frac{5557}{129630} a^{11} - \frac{1679}{51852} a^{10} + \frac{10523}{51852} a^{9} + \frac{48017}{259260} a^{8} + \frac{106867}{259260} a^{7} - \frac{5091}{17284} a^{6} + \frac{16391}{86420} a^{5} + \frac{8683}{25926} a^{4} - \frac{66587}{259260} a^{3} - \frac{19617}{86420} a^{2} + \frac{99733}{259260} a + \frac{913}{2235}$, $\frac{1}{7144815059150776069717353900} a^{15} - \frac{3177824898982281887977}{2381605019716925356572451300} a^{14} + \frac{32661543653769191215831}{2381605019716925356572451300} a^{13} - \frac{227480737479672623658074}{595401254929231339143112825} a^{12} + \frac{33334431755509721616859243}{7144815059150776069717353900} a^{11} - \frac{24630251457750015898499139}{595401254929231339143112825} a^{10} - \frac{1899593826404732270533331}{476321003943385071314490260} a^{9} + \frac{7979145492595558943933879}{82124311024721564019739700} a^{8} - \frac{456934757213455082393661133}{1428963011830155213943470780} a^{7} + \frac{998655854590016192084500993}{2381605019716925356572451300} a^{6} - \frac{544058097145782450612992899}{3572407529575388034858676950} a^{5} - \frac{356104756943062141260583279}{714481505915077606971735390} a^{4} + \frac{287395586405328971357657269}{1786203764787694017429338475} a^{3} + \frac{435964302515928406487002187}{1428963011830155213943470780} a^{2} + \frac{271577043383761270151831293}{595401254929231339143112825} a + \frac{1673883843180029906407393}{7947513970134344905136100}$

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:  $7$
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:  \( 14811624.1093 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{745}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{5}) \), 4.0.3725.1 x2, \(\Q(\sqrt{5}, \sqrt{149})\), 4.0.111005.1 x2, 8.0.308052750625.2 x2, 8.4.45899859843125.1 x2, 8.0.308052750625.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149Data not computed