Properties

Label 16.0.4540317078515625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 11^{6}$
Root discriminant $9.52$
Ramified primes $3, 5, 11$
Class number $1$
Class group Trivial
Galois Group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 4, -3, 0, 0, 11, -24, 29, -24, 11, 0, 0, -3, 4, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*x + 1)
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 3 x^{15} \) \(\mathstrut +\mathstrut 4 x^{14} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut +\mathstrut 11 x^{10} \) \(\mathstrut -\mathstrut 24 x^{9} \) \(\mathstrut +\mathstrut 29 x^{8} \) \(\mathstrut -\mathstrut 24 x^{7} \) \(\mathstrut +\mathstrut 11 x^{6} \) \(\mathstrut -\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(4540317078515625=3^{8}\cdot 5^{8}\cdot 11^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.52$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{163} a^{14} - \frac{43}{163} a^{13} - \frac{70}{163} a^{12} + \frac{69}{163} a^{11} + \frac{81}{163} a^{10} - \frac{49}{163} a^{9} - \frac{66}{163} a^{8} + \frac{57}{163} a^{7} - \frac{66}{163} a^{6} - \frac{49}{163} a^{5} + \frac{81}{163} a^{4} + \frac{69}{163} a^{3} - \frac{70}{163} a^{2} - \frac{43}{163} a + \frac{1}{163}$, $\frac{1}{1793} a^{15} - \frac{4}{1793} a^{14} - \frac{443}{1793} a^{13} + \frac{10}{163} a^{12} - \frac{74}{163} a^{11} + \frac{16}{163} a^{10} - \frac{76}{163} a^{9} - \frac{398}{1793} a^{8} - \frac{288}{1793} a^{7} - \frac{31}{163} a^{6} - \frac{33}{163} a^{5} - \frac{77}{163} a^{4} + \frac{16}{163} a^{3} - \frac{817}{1793} a^{2} + \frac{117}{1793} a - \frac{450}{1793}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{2135}{1793} a^{15} - \frac{4745}{1793} a^{14} + \frac{4463}{1793} a^{13} - \frac{192}{163} a^{12} - \frac{199}{163} a^{11} - \frac{161}{163} a^{10} + \frac{2091}{163} a^{9} - \frac{33365}{1793} a^{8} + \frac{31755}{1793} a^{7} - \frac{1913}{163} a^{6} + \frac{334}{163} a^{5} + \frac{307}{163} a^{4} + \frac{589}{163} a^{3} - \frac{5371}{1793} a^{2} + \frac{4132}{1793} a - \frac{1286}{1793} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2345}{1793} a^{15} - \frac{4859}{1793} a^{14} + \frac{3932}{1793} a^{13} - \frac{104}{163} a^{12} - \frac{264}{163} a^{11} - \frac{257}{163} a^{10} + \frac{2294}{163} a^{9} - \frac{32179}{1793} a^{8} + \frac{27001}{1793} a^{7} - \frac{1369}{163} a^{6} + \frac{113}{163} a^{5} + \frac{241}{163} a^{4} + \frac{516}{163} a^{3} - \frac{5429}{1793} a^{2} + \frac{2863}{1793} a - \frac{31}{1793} \),  \( \frac{3079}{1793} a^{15} - \frac{7102}{1793} a^{14} + \frac{7571}{1793} a^{13} - \frac{434}{163} a^{12} - \frac{192}{163} a^{11} - \frac{199}{163} a^{10} + \frac{2918}{163} a^{9} - \frac{50895}{1793} a^{8} + \frac{55926}{1793} a^{7} - \frac{3831}{163} a^{6} + \frac{1166}{163} a^{5} + \frac{334}{163} a^{4} + \frac{307}{163} a^{3} - \frac{2758}{1793} a^{2} + \frac{6945}{1793} a - \frac{5105}{1793} \),  \( \frac{2560}{1793} a^{15} - \frac{7281}{1793} a^{14} + \frac{8127}{1793} a^{13} - \frac{402}{163} a^{12} - \frac{218}{163} a^{11} - \frac{6}{163} a^{10} + \frac{2692}{163} a^{9} - \frac{55896}{1793} a^{8} + \frac{55347}{1793} a^{7} - \frac{3552}{163} a^{6} + \frac{954}{163} a^{5} + \frac{546}{163} a^{4} + \frac{189}{163} a^{3} - \frac{10782}{1793} a^{2} + \frac{7327}{1793} a - \frac{5107}{1793} \),  \( \frac{3991}{1793} a^{15} - \frac{10046}{1793} a^{14} + \frac{10775}{1793} a^{13} - \frac{521}{163} a^{12} - \frac{346}{163} a^{11} - \frac{146}{163} a^{10} + \frac{3983}{163} a^{9} - \frac{74840}{1793} a^{8} + \frac{75453}{1793} a^{7} - \frac{4705}{163} a^{6} + \frac{1186}{163} a^{5} + \frac{820}{163} a^{4} + \frac{407}{163} a^{3} - \frac{11808}{1793} a^{2} + \frac{11657}{1793} a - \frac{5997}{1793} \),  \( \frac{1885}{1793} a^{15} - \frac{4438}{1793} a^{14} + \frac{3365}{1793} a^{13} - \frac{75}{163} a^{12} - \frac{227}{163} a^{11} - \frac{136}{163} a^{10} + \frac{2010}{163} a^{9} - \frac{29774}{1793} a^{8} + \frac{21222}{1793} a^{7} - \frac{1252}{163} a^{6} - \frac{65}{163} a^{5} + \frac{273}{163} a^{4} + \frac{555}{163} a^{3} - \frac{5424}{1793} a^{2} + \frac{1095}{1793} a - \frac{2438}{1793} \),  \( \frac{3079}{1793} a^{15} - \frac{7102}{1793} a^{14} + \frac{7571}{1793} a^{13} - \frac{434}{163} a^{12} - \frac{192}{163} a^{11} - \frac{199}{163} a^{10} + \frac{2918}{163} a^{9} - \frac{50895}{1793} a^{8} + \frac{55926}{1793} a^{7} - \frac{3831}{163} a^{6} + \frac{1166}{163} a^{5} + \frac{334}{163} a^{4} + \frac{307}{163} a^{3} - \frac{4551}{1793} a^{2} + \frac{6945}{1793} a - \frac{5105}{1793} \),  \( \frac{3009}{1793} a^{15} - \frac{7526}{1793} a^{14} + \frac{7891}{1793} a^{13} - \frac{403}{163} a^{12} - \frac{243}{163} a^{11} - \frac{146}{163} a^{10} + \frac{3061}{163} a^{9} - \frac{55463}{1793} a^{8} + \frac{55681}{1793} a^{7} - \frac{3631}{163} a^{6} + \frac{907}{163} a^{5} + \frac{214}{163} a^{4} + \frac{639}{163} a^{3} - \frac{9247}{1793} a^{2} + \frac{7511}{1793} a - \frac{4790}{1793} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 27.6507723749 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.2.67381875.1 x2, 8.2.7486875.1 x2, 8.0.6125625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$