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Properties

Label 16.0.21614403506...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 19^{12}$
Root discriminant $24.88$
Ramified primes $5, 19$
Class number $4$ (GRH)
Class group $[4]$ (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![8105, -21280, 32730, -37120, 27126, -11022, 1800, 934, -687, -32, 8, 124, -79, 12, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 12*x^13 - 79*x^12 + 124*x^11 + 8*x^10 - 32*x^9 - 687*x^8 + 934*x^7 + 1800*x^6 - 11022*x^5 + 27126*x^4 - 37120*x^3 + 32730*x^2 - 21280*x + 8105)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 + 12*x^13 - 79*x^12 + 124*x^11 + 8*x^10 - 32*x^9 - 687*x^8 + 934*x^7 + 1800*x^6 - 11022*x^5 + 27126*x^4 - 37120*x^3 + 32730*x^2 - 21280*x + 8105, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} + 12 x^{13} - 79 x^{12} + 124 x^{11} + 8 x^{10} - 32 x^{9} - 687 x^{8} + 934 x^{7} + 1800 x^{6} - 11022 x^{5} + 27126 x^{4} - 37120 x^{3} + 32730 x^{2} - 21280 x + 8105 \)

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:  \(21614403506505478515625=5^{10}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.88$
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:  $5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{30} a^{12} - \frac{7}{30} a^{11} + \frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{13}{30} a^{5} - \frac{7}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{30} a^{13} - \frac{1}{15} a^{11} - \frac{1}{6} a^{10} - \frac{2}{15} a^{9} - \frac{1}{10} a^{8} + \frac{11}{30} a^{6} + \frac{1}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{130350} a^{14} + \frac{491}{130350} a^{13} + \frac{416}{65175} a^{12} + \frac{2043}{8690} a^{11} - \frac{10603}{65175} a^{10} - \frac{28553}{130350} a^{9} - \frac{92}{1975} a^{8} + \frac{40427}{130350} a^{7} - \frac{35861}{130350} a^{6} - \frac{2639}{26070} a^{5} + \frac{9017}{43450} a^{4} + \frac{4552}{13035} a^{3} - \frac{1409}{13035} a^{2} + \frac{7901}{26070} a - \frac{3647}{13035}$, $\frac{1}{53970982206478959591750} a^{15} + \frac{80868992297963}{30928929631220034150} a^{14} + \frac{798105772835653011671}{53970982206478959591750} a^{13} - \frac{63438680614708757777}{53970982206478959591750} a^{12} - \frac{1421590336693972107068}{26985491103239479795875} a^{11} - \frac{9622832855417292079577}{53970982206478959591750} a^{10} + \frac{290677884272597004197}{1635484309287241199750} a^{9} + \frac{4453779061973751397867}{26985491103239479795875} a^{8} - \frac{2055327190435562552009}{26985491103239479795875} a^{7} - \frac{380733757100447537152}{26985491103239479795875} a^{6} + \frac{7942406390014929139661}{53970982206478959591750} a^{5} - \frac{4511666351831286533207}{17990327402159653197250} a^{4} + \frac{1604751263029832140574}{5397098220647895959175} a^{3} - \frac{1965620353573501572853}{5397098220647895959175} a^{2} + \frac{92096860534730552849}{431767857651831676734} a + \frac{33695490194903085734}{163548430928724119975}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 25802.5997037 \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), 4.0.1805.1 x2, 4.2.475.1 x2, \(\Q(\sqrt{5}, \sqrt{-19})\), 8.0.147018378125.1 x2, 8.0.29403675625.1 x2, 8.0.81450625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$