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Properties

Label 16.8.66688975910...2433.2
Degree $16$
Signature $[8, 4]$
Discriminant $13^{12}\cdot 17^{15}$
Root discriminant $97.50$
Ramified primes $13, 17$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1672427, 2558448, -2215830, -1186125, 2075106, -449804, -55402, -83318, 95150, -50814, 11918, 1070, -1461, 526, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 84*x^14 + 526*x^13 - 1461*x^12 + 1070*x^11 + 11918*x^10 - 50814*x^9 + 95150*x^8 - 83318*x^7 - 55402*x^6 - 449804*x^5 + 2075106*x^4 - 1186125*x^3 - 2215830*x^2 + 2558448*x + 1672427)
 
gp: K = bnfinit(x^16 - x^15 - 84*x^14 + 526*x^13 - 1461*x^12 + 1070*x^11 + 11918*x^10 - 50814*x^9 + 95150*x^8 - 83318*x^7 - 55402*x^6 - 449804*x^5 + 2075106*x^4 - 1186125*x^3 - 2215830*x^2 + 2558448*x + 1672427, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 84 x^{14} + 526 x^{13} - 1461 x^{12} + 1070 x^{11} + 11918 x^{10} - 50814 x^{9} + 95150 x^{8} - 83318 x^{7} - 55402 x^{6} - 449804 x^{5} + 2075106 x^{4} - 1186125 x^{3} - 2215830 x^{2} + 2558448 x + 1672427 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(66688975910627504451630153142433=13^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.50$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1313} a^{14} - \frac{42}{101} a^{13} - \frac{406}{1313} a^{12} - \frac{257}{1313} a^{11} + \frac{522}{1313} a^{10} + \frac{28}{1313} a^{9} - \frac{437}{1313} a^{8} + \frac{105}{1313} a^{7} - \frac{46}{1313} a^{6} + \frac{463}{1313} a^{5} + \frac{69}{1313} a^{4} - \frac{197}{1313} a^{3} - \frac{581}{1313} a^{2} - \frac{85}{1313} a + \frac{415}{1313}$, $\frac{1}{12811144637189090497649703918226549394870389432283} a^{15} - \frac{170157857880304532617486518746362039290156258}{12811144637189090497649703918226549394870389432283} a^{14} + \frac{4575781621444035591783766284117318592969990229332}{12811144637189090497649703918226549394870389432283} a^{13} + \frac{753814575983515153737083405467955217382782149997}{12811144637189090497649703918226549394870389432283} a^{12} + \frac{3216426403789164364529938519573323717388309581791}{12811144637189090497649703918226549394870389432283} a^{11} + \frac{193109925707763492744854002448967989007262802824}{12811144637189090497649703918226549394870389432283} a^{10} - \frac{3519246188476721932409061065789455288435586789253}{12811144637189090497649703918226549394870389432283} a^{9} + \frac{4032910241733586388979090140472574552945410530974}{12811144637189090497649703918226549394870389432283} a^{8} + \frac{717151101642894575153072532591013427476345313399}{12811144637189090497649703918226549394870389432283} a^{7} + \frac{2320557495268775135704456780618883086545848015601}{12811144637189090497649703918226549394870389432283} a^{6} - \frac{5212104078278751538666254505843253704137230376393}{12811144637189090497649703918226549394870389432283} a^{5} - \frac{2288200436480687773873671874911183104062340180090}{12811144637189090497649703918226549394870389432283} a^{4} - \frac{199854336906023692432269200037996210327542247618}{12811144637189090497649703918226549394870389432283} a^{3} - \frac{802294395171693931657305354093858542317869230938}{12811144637189090497649703918226549394870389432283} a^{2} + \frac{4859692558649496426984395501704596766313520783031}{12811144637189090497649703918226549394870389432283} a + \frac{4001615257469664439577230793118507875313873365585}{12811144637189090497649703918226549394870389432283}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $11$
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:  \( 1449358615.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.11719682839553.1

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

Sibling fields

Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
17Data not computed