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Properties

Label 16.0.34984505969...9921.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 37^{12}$
Root discriminant $34.20$
Ramified primes $3, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, -6561, 3645, 729, -1701, 891, 153, -315, 301, -105, 17, 33, -21, 3, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 + 3*x^13 - 21*x^12 + 33*x^11 + 17*x^10 - 105*x^9 + 301*x^8 - 315*x^7 + 153*x^6 + 891*x^5 - 1701*x^4 + 729*x^3 + 3645*x^2 - 6561*x + 6561)
 
gp: K = bnfinit(x^16 - 3*x^15 + 5*x^14 + 3*x^13 - 21*x^12 + 33*x^11 + 17*x^10 - 105*x^9 + 301*x^8 - 315*x^7 + 153*x^6 + 891*x^5 - 1701*x^4 + 729*x^3 + 3645*x^2 - 6561*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 5 x^{14} + 3 x^{13} - 21 x^{12} + 33 x^{11} + 17 x^{10} - 105 x^{9} + 301 x^{8} - 315 x^{7} + 153 x^{6} + 891 x^{5} - 1701 x^{4} + 729 x^{3} + 3645 x^{2} - 6561 x + 6561 \)

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:  \(3498450596935634189769921=3^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.20$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{4}{9} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{4}{27} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{1}{9} a^{6} + \frac{8}{27} a^{5} + \frac{13}{27} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{2997} a^{12} + \frac{16}{999} a^{11} + \frac{104}{2997} a^{10} - \frac{22}{999} a^{9} + \frac{248}{999} a^{8} - \frac{211}{999} a^{7} - \frac{1081}{2997} a^{6} + \frac{218}{999} a^{5} + \frac{850}{2997} a^{4} + \frac{368}{999} a^{3} - \frac{7}{37} a^{2} - \frac{49}{111} a + \frac{9}{37}$, $\frac{1}{8991} a^{13} + \frac{131}{8991} a^{11} - \frac{7}{999} a^{10} + \frac{194}{2997} a^{9} - \frac{793}{2997} a^{8} - \frac{1666}{8991} a^{7} - \frac{89}{333} a^{6} + \frac{94}{8991} a^{5} - \frac{911}{2997} a^{4} + \frac{413}{999} a^{3} + \frac{71}{333} a^{2} + \frac{53}{111} a + \frac{4}{37}$, $\frac{1}{26973} a^{14} - \frac{4}{26973} a^{12} + \frac{50}{2997} a^{11} - \frac{490}{8991} a^{10} - \frac{1153}{8991} a^{9} + \frac{2789}{26973} a^{8} + \frac{301}{999} a^{7} + \frac{8167}{26973} a^{6} + \frac{3292}{8991} a^{5} + \frac{317}{2997} a^{4} - \frac{28}{333} a^{3} - \frac{38}{333} a^{2} - \frac{38}{111} a + \frac{13}{37}$, $\frac{1}{32934033} a^{15} - \frac{80}{10978011} a^{14} - \frac{1723}{32934033} a^{13} + \frac{587}{10978011} a^{12} - \frac{137905}{10978011} a^{11} + \frac{5873}{10978011} a^{10} - \frac{4230631}{32934033} a^{9} - \frac{2567395}{10978011} a^{8} + \frac{9456592}{32934033} a^{7} + \frac{1712261}{10978011} a^{6} + \frac{1496422}{3659337} a^{5} - \frac{15551}{1219779} a^{4} + \frac{5961}{15059} a^{3} - \frac{4401}{15059} a^{2} - \frac{4308}{15059} a - \frac{7496}{15059}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -1 \) (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:  \( 1108264.92903 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.4.455877.1, 8.2.623471517387.2 x2, 8.4.1870414552161.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$