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Properties

Label 16.16.7921586895...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 19^{8}\cdot 103^{6}$
Root discriminant $55.42$
Ramified primes $5, 19, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![821, -9338, 19997, 69777, -244557, 184877, 81692, -148144, 24297, 31709, -10417, -2708, 1231, 93, -59, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 59*x^14 + 93*x^13 + 1231*x^12 - 2708*x^11 - 10417*x^10 + 31709*x^9 + 24297*x^8 - 148144*x^7 + 81692*x^6 + 184877*x^5 - 244557*x^4 + 69777*x^3 + 19997*x^2 - 9338*x + 821)
 
gp: K = bnfinit(x^16 - x^15 - 59*x^14 + 93*x^13 + 1231*x^12 - 2708*x^11 - 10417*x^10 + 31709*x^9 + 24297*x^8 - 148144*x^7 + 81692*x^6 + 184877*x^5 - 244557*x^4 + 69777*x^3 + 19997*x^2 - 9338*x + 821, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 59 x^{14} + 93 x^{13} + 1231 x^{12} - 2708 x^{11} - 10417 x^{10} + 31709 x^{9} + 24297 x^{8} - 148144 x^{7} + 81692 x^{6} + 184877 x^{5} - 244557 x^{4} + 69777 x^{3} + 19997 x^{2} - 9338 x + 821 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7921586895449647259644140625=5^{8}\cdot 19^{8}\cdot 103^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.42$
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:  $5, 19, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{2}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{9}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{2}{27} a^{11} + \frac{7}{27} a^{10} + \frac{4}{9} a^{9} - \frac{10}{27} a^{8} + \frac{11}{27} a^{6} - \frac{10}{27} a^{5} + \frac{11}{27} a^{4} - \frac{8}{27} a^{3} + \frac{7}{27} a^{2} - \frac{10}{27} a - \frac{8}{27}$, $\frac{1}{1377} a^{14} - \frac{2}{1377} a^{13} - \frac{28}{1377} a^{12} + \frac{7}{153} a^{11} + \frac{356}{1377} a^{10} + \frac{140}{1377} a^{9} - \frac{233}{1377} a^{8} + \frac{200}{1377} a^{7} - \frac{11}{27} a^{6} + \frac{16}{459} a^{5} + \frac{656}{1377} a^{4} - \frac{94}{459} a^{3} + \frac{577}{1377} a^{2} + \frac{434}{1377} a + \frac{494}{1377}$, $\frac{1}{12110299291023731404205193} a^{15} + \frac{369148980574325738443}{4036766430341243801401731} a^{14} + \frac{39715704675138986541421}{12110299291023731404205193} a^{13} - \frac{536383587746190725885579}{12110299291023731404205193} a^{12} + \frac{232592907696228227330804}{1730042755860533057743599} a^{11} - \frac{23194970741834630193934}{237456848843602576553043} a^{10} + \frac{3273901586526039288777710}{12110299291023731404205193} a^{9} + \frac{401124622867246285499014}{12110299291023731404205193} a^{8} - \frac{3106815318138647172137447}{12110299291023731404205193} a^{7} - \frac{1997066364394228147436170}{4036766430341243801401731} a^{6} - \frac{1573936544744898463440049}{12110299291023731404205193} a^{5} - \frac{204862983277636317712049}{12110299291023731404205193} a^{4} + \frac{744602046589348793605186}{1730042755860533057743599} a^{3} + \frac{2525713024913597271997672}{12110299291023731404205193} a^{2} - \frac{99371684674800244387817}{448529603371249311266859} a + \frac{4192254737902047745842313}{12110299291023731404205193}$

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:  $15$
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:  \( 616122428.353 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.2.2$x^{4} - 103 x^{2} + 53045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$