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Properties

Label 16.0.37909967031...8041.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{10}$
Root discriminant $29.76$
Ramified primes $3, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![813, 375, 16878, 2622, -17162, -1283, 7286, -2124, -1877, 1253, 244, -256, 85, 42, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813)
 
gp: K = bnfinit(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 16 x^{14} + 42 x^{13} + 85 x^{12} - 256 x^{11} + 244 x^{10} + 1253 x^{9} - 1877 x^{8} - 2124 x^{7} + 7286 x^{6} - 1283 x^{5} - 17162 x^{4} + 2622 x^{3} + 16878 x^{2} + 375 x + 813 \)

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:  \(379099670317033992358041=3^{12}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.76$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{15} a^{12} - \frac{4}{15} a^{10} - \frac{1}{5} a^{8} - \frac{1}{15} a^{6} - \frac{2}{15} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{45} a^{13} + \frac{1}{45} a^{12} + \frac{2}{45} a^{11} - \frac{13}{45} a^{10} - \frac{7}{15} a^{9} + \frac{1}{5} a^{8} - \frac{7}{45} a^{7} - \frac{7}{45} a^{6} - \frac{14}{45} a^{5} + \frac{16}{45} a^{4} - \frac{2}{5} a^{3} + \frac{4}{15} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{22995} a^{14} + \frac{26}{22995} a^{13} - \frac{1}{7665} a^{12} + \frac{1297}{22995} a^{11} + \frac{1754}{22995} a^{10} - \frac{3607}{7665} a^{9} - \frac{2392}{22995} a^{8} + \frac{9358}{22995} a^{7} - \frac{848}{7665} a^{6} - \frac{73}{315} a^{5} + \frac{7597}{22995} a^{4} - \frac{263}{1095} a^{3} + \frac{8}{73} a^{2} + \frac{160}{511} a - \frac{391}{1533}$, $\frac{1}{67564767053753701673559975} a^{15} - \frac{481579852261049588138}{67564767053753701673559975} a^{14} + \frac{201520679984963353541264}{22521589017917900557853325} a^{13} - \frac{3026344513462114197797}{900863560716716022314133} a^{12} - \frac{104759613568971896860033}{2702590682150148066942399} a^{11} + \frac{29441865172651675626813089}{67564767053753701673559975} a^{10} - \frac{5526963548046097005184796}{13512953410750740334711995} a^{9} - \frac{14087878057689079042003177}{67564767053753701673559975} a^{8} - \frac{116165245401882504831553}{4504317803583580111570665} a^{7} - \frac{11216872618509812388309158}{22521589017917900557853325} a^{6} - \frac{325558997367962134564888}{1930421915821534333530285} a^{5} - \frac{7067952353409911111763368}{67564767053753701673559975} a^{4} - \frac{951443901536924675677819}{3217369859702557222550475} a^{3} - \frac{8739990991931902165242823}{22521589017917900557853325} a^{2} + \frac{7989476411264126657409794}{22521589017917900557853325} a - \frac{4306292872655061552621464}{22521589017917900557853325}$

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:  \( -\frac{36921365106}{188280668324725} a^{15} + \frac{118310705419}{564842004974175} a^{14} + \frac{2022846387779}{564842004974175} a^{13} - \frac{219281853057}{37656133664945} a^{12} - \frac{2807885611939}{112968400994835} a^{11} + \frac{24458951411648}{564842004974175} a^{10} - \frac{73655926932}{7531226732989} a^{9} - \frac{188538992666614}{564842004974175} a^{8} + \frac{27749547249101}{112968400994835} a^{7} + \frac{135873862258019}{188280668324725} a^{6} - \frac{174735924804509}{112968400994835} a^{5} - \frac{328815252097976}{564842004974175} a^{4} + \frac{786609814496154}{188280668324725} a^{3} + \frac{76533010158734}{188280668324725} a^{2} - \frac{722082103992477}{188280668324725} a + \frac{92221951777917}{188280668324725} \) (order $6$)
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:  \( 435198.818124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-183}) \), 4.2.11163.1 x2, 4.0.549.1 x2, \(\Q(\sqrt{-3}, \sqrt{61})\), 8.0.615710703429.2, 8.0.165469149.1, 8.0.1121513121.2

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 sibling: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data not computed