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Properties

Label 16.0.78929890744...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}\cdot 19^{6}$
Root discriminant $47.98$
Ramified primes $2, 5, 19$
Class number $156$ (GRH)
Class group $[2, 78]$ (GRH)
Galois group $(C_2\times C_8):C_2^2$ (as 16T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![364816, 0, -64768, 0, 69296, 0, -6704, 0, 2204, 0, 8, 0, 104, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 104*x^12 + 8*x^10 + 2204*x^8 - 6704*x^6 + 69296*x^4 - 64768*x^2 + 364816)
 
gp: K = bnfinit(x^16 + 4*x^14 + 104*x^12 + 8*x^10 + 2204*x^8 - 6704*x^6 + 69296*x^4 - 64768*x^2 + 364816, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 104 x^{12} + 8 x^{10} + 2204 x^{8} - 6704 x^{6} + 69296 x^{4} - 64768 x^{2} + 364816 \)

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:  \(789298907447296000000000000=2^{36}\cdot 5^{12}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.98$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{48} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{5}{12}$, $\frac{1}{48} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a$, $\frac{1}{48} a^{10} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2}$, $\frac{1}{48} a^{11} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{5}{12} a^{3} - \frac{1}{2} a$, $\frac{1}{288} a^{12} + \frac{1}{144} a^{10} + \frac{1}{144} a^{8} + \frac{1}{12} a^{6} - \frac{5}{72} a^{4} - \frac{1}{2} a^{3} + \frac{4}{9} a^{2} - \frac{1}{2} a - \frac{2}{9}$, $\frac{1}{288} a^{13} + \frac{1}{144} a^{11} + \frac{1}{144} a^{9} + \frac{1}{12} a^{7} - \frac{5}{72} a^{5} + \frac{4}{9} a^{3} - \frac{1}{2} a^{2} - \frac{2}{9} a$, $\frac{1}{4868988940512} a^{14} + \frac{7364343839}{4868988940512} a^{12} + \frac{677829700}{152155904391} a^{10} + \frac{192468791}{135249692792} a^{8} - \frac{1}{4} a^{7} - \frac{21731549953}{152155904391} a^{6} - \frac{1}{4} a^{5} - \frac{25324107632}{152155904391} a^{4} + \frac{21170123662}{152155904391} a^{2} - \frac{62436229747}{202874539188}$, $\frac{1}{735217330017312} a^{15} + \frac{49525901327}{91902166252164} a^{13} + \frac{514629297703}{61268110834776} a^{11} + \frac{9319215359}{91902166252164} a^{9} + \frac{7165838576159}{91902166252164} a^{7} + \frac{20608812778247}{91902166252164} a^{5} - \frac{4816375233687}{10211351805796} a^{3} + \frac{2738471188625}{22975541563041} a$

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

Class group and class number

$C_{2}\times C_{78}$, which has order $156$ (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:  \( 42940.0557728 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8):C_2^2$ (as 16T75):

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 $(C_2\times C_8):C_2^2$
Character table for $(C_2\times C_8):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.7600.1, \(\Q(\zeta_{20})^+\), 4.4.38000.1, 8.0.7023616000000.3, 8.0.280944640000.1, 8.8.23104000000.1

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
Arithmetically equvalently 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$