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Properties

Label 16.0.29476538281...9968.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 193^{5}$
Root discriminant $25.37$
Ramified primes $2, 3, 193$
Class number $2$
Class group $[2]$
Galois group 16T1163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16749, 59688, 95484, 60252, -6776, -18808, -2772, -364, -246, 1048, 36, -332, 48, 56, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 12*x^14 + 56*x^13 + 48*x^12 - 332*x^11 + 36*x^10 + 1048*x^9 - 246*x^8 - 364*x^7 - 2772*x^6 - 18808*x^5 - 6776*x^4 + 60252*x^3 + 95484*x^2 + 59688*x + 16749)
 
gp: K = bnfinit(x^16 - 4*x^15 - 12*x^14 + 56*x^13 + 48*x^12 - 332*x^11 + 36*x^10 + 1048*x^9 - 246*x^8 - 364*x^7 - 2772*x^6 - 18808*x^5 - 6776*x^4 + 60252*x^3 + 95484*x^2 + 59688*x + 16749, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 12 x^{14} + 56 x^{13} + 48 x^{12} - 332 x^{11} + 36 x^{10} + 1048 x^{9} - 246 x^{8} - 364 x^{7} - 2772 x^{6} - 18808 x^{5} - 6776 x^{4} + 60252 x^{3} + 95484 x^{2} + 59688 x + 16749 \)

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:  \(29476538281722504019968=2^{24}\cdot 3^{8}\cdot 193^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.37$
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:  $2, 3, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{3}{8} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} + \frac{3}{16} a + \frac{1}{4}$, $\frac{1}{480} a^{12} + \frac{1}{120} a^{11} - \frac{1}{60} a^{10} + \frac{1}{20} a^{9} - \frac{1}{480} a^{8} + \frac{1}{15} a^{7} - \frac{1}{30} a^{6} + \frac{1}{5} a^{5} + \frac{19}{480} a^{4} + \frac{31}{120} a^{3} - \frac{19}{60} a^{2} - \frac{9}{20} a + \frac{23}{160}$, $\frac{1}{1440} a^{13} + \frac{1}{1440} a^{12} - \frac{1}{72} a^{11} - \frac{1}{120} a^{10} - \frac{73}{1440} a^{9} - \frac{5}{288} a^{8} + \frac{1}{180} a^{7} + \frac{1}{10} a^{6} + \frac{331}{1440} a^{5} + \frac{307}{1440} a^{4} - \frac{101}{360} a^{3} + \frac{5}{24} a^{2} + \frac{119}{480} a - \frac{43}{160}$, $\frac{1}{1440} a^{14} + \frac{1}{720} a^{11} + \frac{41}{1440} a^{10} - \frac{13}{240} a^{9} - \frac{13}{240} a^{8} - \frac{23}{360} a^{7} + \frac{31}{1440} a^{6} - \frac{29}{120} a^{5} + \frac{19}{120} a^{4} + \frac{169}{720} a^{3} + \frac{25}{96} a^{2} - \frac{17}{48} a + \frac{17}{80}$, $\frac{1}{18119465202875040} a^{15} + \frac{5455166104819}{18119465202875040} a^{14} + \frac{223258445527}{6039821734291680} a^{13} + \frac{1080937165699}{9059732601437520} a^{12} - \frac{362616042489131}{18119465202875040} a^{11} - \frac{72637124060551}{18119465202875040} a^{10} + \frac{259910948327899}{6039821734291680} a^{9} - \frac{518022534584677}{9059732601437520} a^{8} + \frac{723466529584751}{18119465202875040} a^{7} + \frac{136406443331909}{3623893040575008} a^{6} + \frac{403075892657483}{2013273911430560} a^{5} - \frac{1400271427355819}{9059732601437520} a^{4} - \frac{5035753897689157}{18119465202875040} a^{3} - \frac{318380208262115}{1207964346858336} a^{2} + \frac{1514643445131421}{6039821734291680} a - \frac{421753334162411}{1006636955715280}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{1234577793457}{1509955433572920} a^{15} + \frac{3834636220967}{905973260143752} a^{14} + \frac{42984857125717}{9059732601437520} a^{13} - \frac{115373874314141}{2264933150359380} a^{12} + \frac{19098354198593}{905973260143752} a^{11} + \frac{136907836945693}{566233287589845} a^{10} - \frac{2801352225600253}{9059732601437520} a^{9} - \frac{269115237057409}{566233287589845} a^{8} + \frac{652216914605011}{905973260143752} a^{7} - \frac{2520065042445797}{4529866300718760} a^{6} + \frac{26874356939105599}{9059732601437520} a^{5} + \frac{5312166622741877}{452986630071876} a^{4} - \frac{36102036636466553}{4529866300718760} a^{3} - \frac{29583331455702823}{754977716786460} a^{2} - \frac{98139273253264517}{3019910867145840} a - \frac{706724704748523}{62914809732205} \) (order $12$)
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:  \( 176192.924057 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1163 are not computed
Character table for t16n1163 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.4002048.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
193Data not computed