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Properties

Label 16.8.77176147591...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{12}\cdot 5^{12}\cdot 29^{6}$
Root discriminant $26.94$
Ramified primes $3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T322)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 120, -60, -718, -229, 930, 294, -441, -54, -82, -3, 96, -18, 7, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 7*x^13 - 18*x^12 + 96*x^11 - 3*x^10 - 82*x^9 - 54*x^8 - 441*x^7 + 294*x^6 + 930*x^5 - 229*x^4 - 718*x^3 - 60*x^2 + 120*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 7*x^13 - 18*x^12 + 96*x^11 - 3*x^10 - 82*x^9 - 54*x^8 - 441*x^7 + 294*x^6 + 930*x^5 - 229*x^4 - 718*x^3 - 60*x^2 + 120*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 7 x^{13} - 18 x^{12} + 96 x^{11} - 3 x^{10} - 82 x^{9} - 54 x^{8} - 441 x^{7} + 294 x^{6} + 930 x^{5} - 229 x^{4} - 718 x^{3} - 60 x^{2} + 120 x + 16 \)

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:  \(77176147591689697265625=3^{12}\cdot 5^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.94$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{22298831013730193368} a^{15} + \frac{15739886571318016}{146702835616646009} a^{14} + \frac{2517266664245615185}{11149415506865096684} a^{13} - \frac{6225061250201792101}{22298831013730193368} a^{12} - \frac{1977440298525505981}{5574707753432548342} a^{11} - \frac{1049322486911117393}{2787353876716274171} a^{10} - \frac{4681314611831146915}{22298831013730193368} a^{9} + \frac{783472367134052110}{2787353876716274171} a^{8} - \frac{4108924409425378651}{11149415506865096684} a^{7} - \frac{2782930778541158317}{22298831013730193368} a^{6} - \frac{128308900711037429}{293405671233292018} a^{5} - \frac{4242527798015894227}{11149415506865096684} a^{4} - \frac{8005181894597985}{22298831013730193368} a^{3} - \frac{1379975072987138202}{2787353876716274171} a^{2} + \frac{2193218588299175775}{5574707753432548342} a + \frac{211020228542385832}{2787353876716274171}$

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:  $11$
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:  \( 311894.804498 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T322):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.32625.1, 4.4.725.1, 8.8.1064390625.1

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

Sibling fields

Degree 16 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$