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Properties

Label 16.0.10236206731...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 109^{6}$
Root discriminant $23.75$
Ramified primes $5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![661, 5684, 13879, 7394, 18806, 2282, 9681, -238, 2867, -662, 781, -302, 166, -54, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 19*x^14 - 54*x^13 + 166*x^12 - 302*x^11 + 781*x^10 - 662*x^9 + 2867*x^8 - 238*x^7 + 9681*x^6 + 2282*x^5 + 18806*x^4 + 7394*x^3 + 13879*x^2 + 5684*x + 661)
 
gp: K = bnfinit(x^16 - 4*x^15 + 19*x^14 - 54*x^13 + 166*x^12 - 302*x^11 + 781*x^10 - 662*x^9 + 2867*x^8 - 238*x^7 + 9681*x^6 + 2282*x^5 + 18806*x^4 + 7394*x^3 + 13879*x^2 + 5684*x + 661, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 19 x^{14} - 54 x^{13} + 166 x^{12} - 302 x^{11} + 781 x^{10} - 662 x^{9} + 2867 x^{8} - 238 x^{7} + 9681 x^{6} + 2282 x^{5} + 18806 x^{4} + 7394 x^{3} + 13879 x^{2} + 5684 x + 661 \)

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:  \(10236206731207275390625=5^{14}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.75$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{66544037113828133729007940558} a^{15} + \frac{7223616124192309790222385309}{66544037113828133729007940558} a^{14} + \frac{7634150759234913344148639891}{33272018556914066864503970279} a^{13} + \frac{3616514138555839476804106549}{33272018556914066864503970279} a^{12} + \frac{2978527858604198744165829611}{66544037113828133729007940558} a^{11} + \frac{196307942644260862332639359}{1073290921190776350467870009} a^{10} + \frac{7664087339736204924083528989}{66544037113828133729007940558} a^{9} - \frac{7851705095071218586955449255}{33272018556914066864503970279} a^{8} - \frac{1607453029913989270647539402}{33272018556914066864503970279} a^{7} + \frac{12573843127163880283129981385}{66544037113828133729007940558} a^{6} - \frac{14175027056049964459490235551}{33272018556914066864503970279} a^{5} - \frac{26452691904579177743735663213}{66544037113828133729007940558} a^{4} - \frac{1880245776563661198448026451}{33272018556914066864503970279} a^{3} + \frac{7412316913889662274313583577}{33272018556914066864503970279} a^{2} - \frac{13369071391588548274116420535}{66544037113828133729007940558} a - \frac{5561133333510274374338577613}{66544037113828133729007940558}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{12518866406153539791891}{11859568189953329839423978} a^{15} - \frac{50877957715020083089455}{11859568189953329839423978} a^{14} + \frac{247347651793887032073665}{11859568189953329839423978} a^{13} - \frac{362643227361584043583335}{5929784094976664919711989} a^{12} + \frac{2252719454899331015111261}{11859568189953329839423978} a^{11} - \frac{4306928000338632936533431}{11859568189953329839423978} a^{10} + \frac{11095711548747116779747061}{11859568189953329839423978} a^{9} - \frac{5490558481394944439730880}{5929784094976664919711989} a^{8} + \frac{40406519838886017038804531}{11859568189953329839423978} a^{7} - \frac{9829716734866469002211869}{11859568189953329839423978} a^{6} + \frac{65914312000537933541394901}{5929784094976664919711989} a^{5} + \frac{3327735973799626080300975}{5929784094976664919711989} a^{4} + \frac{129202692383050909767756151}{5929784094976664919711989} a^{3} + \frac{19911717006769423244931882}{5929784094976664919711989} a^{2} + \frac{88907668989153504550523361}{5929784094976664919711989} a + \frac{35041555898854613105094455}{11859568189953329839423978} \) (order $10$)
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:  \( 93034.1814672 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.13625.1, \(\Q(\zeta_{5})\), 4.0.2725.1, 8.8.101174140625.1, 8.0.101174140625.1, 8.0.185640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$109$109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.8.6.2$x^{8} + 1199 x^{4} + 427716$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$