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Properties

Label 16.0.13109528905...5649.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 7^{6}\cdot 19^{8}$
Root discriminant $15.66$
Ramified primes $3, 7, 19$
Class number $1$
Class group Trivial
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 27, 42, 93, 64, 37, 115, -45, 153, -88, 102, -50, 42, -12, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 10*x^14 - 12*x^13 + 42*x^12 - 50*x^11 + 102*x^10 - 88*x^9 + 153*x^8 - 45*x^7 + 115*x^6 + 37*x^5 + 64*x^4 + 93*x^3 + 42*x^2 + 27*x + 9)
 
gp: K = bnfinit(x^16 - x^15 + 10*x^14 - 12*x^13 + 42*x^12 - 50*x^11 + 102*x^10 - 88*x^9 + 153*x^8 - 45*x^7 + 115*x^6 + 37*x^5 + 64*x^4 + 93*x^3 + 42*x^2 + 27*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 10 x^{14} - 12 x^{13} + 42 x^{12} - 50 x^{11} + 102 x^{10} - 88 x^{9} + 153 x^{8} - 45 x^{7} + 115 x^{6} + 37 x^{5} + 64 x^{4} + 93 x^{3} + 42 x^{2} + 27 x + 9 \)

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:  \(13109528905069805649=3^{8}\cdot 7^{6}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.66$
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:  $3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{21} a^{12} - \frac{2}{21} a^{11} + \frac{2}{21} a^{10} + \frac{2}{21} a^{9} - \frac{2}{21} a^{8} - \frac{3}{7} a^{7} + \frac{1}{3} a^{6} - \frac{8}{21} a^{5} + \frac{5}{21} a^{4} - \frac{1}{21} a^{3} + \frac{1}{3} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{21} a^{13} - \frac{2}{21} a^{11} - \frac{1}{21} a^{10} - \frac{5}{21} a^{9} + \frac{1}{21} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{4}{21} a^{5} + \frac{2}{21} a^{4} - \frac{3}{7} a^{3} + \frac{1}{21} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{147} a^{14} + \frac{2}{147} a^{13} - \frac{2}{147} a^{12} - \frac{5}{147} a^{11} + \frac{2}{21} a^{10} + \frac{11}{49} a^{9} + \frac{47}{147} a^{8} - \frac{17}{49} a^{7} - \frac{13}{147} a^{6} - \frac{16}{49} a^{5} + \frac{58}{147} a^{4} + \frac{46}{147} a^{3} - \frac{58}{147} a^{2} - \frac{9}{49} a - \frac{8}{49}$, $\frac{1}{51414867} a^{15} + \frac{126209}{51414867} a^{14} - \frac{1026845}{51414867} a^{13} - \frac{154870}{17138289} a^{12} - \frac{693647}{5712763} a^{11} - \frac{435941}{51414867} a^{10} - \frac{4516933}{17138289} a^{9} + \frac{14995922}{51414867} a^{8} + \frac{575723}{2448327} a^{7} - \frac{113093}{5712763} a^{6} - \frac{18125864}{51414867} a^{5} + \frac{3885901}{51414867} a^{4} - \frac{3377081}{7344981} a^{3} - \frac{146633}{2448327} a^{2} - \frac{7439840}{17138289} a + \frac{2386623}{5712763}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{20054}{154399} a^{15} + \frac{36011}{154399} a^{14} - \frac{96073}{66171} a^{13} + \frac{58427}{22057} a^{12} - \frac{3332666}{463197} a^{11} + \frac{5323816}{463197} a^{10} - \frac{9521104}{463197} a^{9} + \frac{11434649}{463197} a^{8} - \frac{15893741}{463197} a^{7} + \frac{12370672}{463197} a^{6} - \frac{12944314}{463197} a^{5} + \frac{1687083}{154399} a^{4} - \frac{5397239}{463197} a^{3} - \frac{800115}{154399} a^{2} - \frac{86361}{154399} a - \frac{261880}{154399} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3179.60910431 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{57}) \), 4.0.1197.2, 4.0.1197.1, \(\Q(\sqrt{-3}, \sqrt{-19})\), 8.0.190563597.4, 8.0.190563597.3, 8.0.517244049.2

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$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}$