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Properties

Label 16.0.43190748110...8641.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 37^{12}$
Root discriminant $25.98$
Ramified primes $3, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -2673, 6399, -8424, 7560, -3132, 567, -207, 372, -213, 28, -41, 53, -20, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 20*x^13 + 53*x^12 - 41*x^11 + 28*x^10 - 213*x^9 + 372*x^8 - 207*x^7 + 567*x^6 - 3132*x^5 + 7560*x^4 - 8424*x^3 + 6399*x^2 - 2673*x + 729)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 20*x^13 + 53*x^12 - 41*x^11 + 28*x^10 - 213*x^9 + 372*x^8 - 207*x^7 + 567*x^6 - 3132*x^5 + 7560*x^4 - 8424*x^3 + 6399*x^2 - 2673*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 20 x^{13} + 53 x^{12} - 41 x^{11} + 28 x^{10} - 213 x^{9} + 372 x^{8} - 207 x^{7} + 567 x^{6} - 3132 x^{5} + 7560 x^{4} - 8424 x^{3} + 6399 x^{2} - 2673 x + 729 \)

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:  \(43190748110316471478641=3^{8}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.98$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{4}{27} a^{7} - \frac{2}{27} a^{6} + \frac{7}{27} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{81} a^{12} + \frac{1}{81} a^{11} - \frac{1}{81} a^{10} - \frac{2}{81} a^{9} - \frac{1}{81} a^{8} + \frac{4}{81} a^{7} + \frac{10}{81} a^{6} + \frac{2}{9} a^{5} - \frac{7}{27} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{891} a^{13} + \frac{2}{891} a^{12} - \frac{1}{297} a^{11} - \frac{14}{297} a^{10} - \frac{1}{33} a^{8} - \frac{136}{891} a^{7} + \frac{34}{891} a^{6} + \frac{127}{297} a^{5} - \frac{67}{297} a^{4} + \frac{46}{99} a^{3} - \frac{47}{99} a^{2} - \frac{7}{33} a - \frac{4}{11}$, $\frac{1}{8019} a^{14} + \frac{1}{8019} a^{13} - \frac{16}{8019} a^{12} - \frac{116}{8019} a^{11} - \frac{310}{8019} a^{10} + \frac{358}{8019} a^{9} + \frac{133}{8019} a^{8} - \frac{167}{2673} a^{7} - \frac{124}{891} a^{6} - \frac{112}{297} a^{5} - \frac{20}{297} a^{4} - \frac{119}{297} a^{3} - \frac{116}{297} a^{2} + \frac{46}{99} a + \frac{16}{33}$, $\frac{1}{2393883097353} a^{15} - \frac{132326522}{2393883097353} a^{14} - \frac{70000576}{2393883097353} a^{13} + \frac{13779339643}{2393883097353} a^{12} + \frac{10505514155}{2393883097353} a^{11} + \frac{125226233656}{2393883097353} a^{10} - \frac{64411298219}{2393883097353} a^{9} - \frac{21865751078}{797961032451} a^{8} + \frac{14402810557}{265987010817} a^{7} - \frac{2149141903}{265987010817} a^{6} + \frac{11679943972}{29554112313} a^{5} - \frac{34372279687}{88662336939} a^{4} - \frac{10146133121}{88662336939} a^{3} + \frac{1471132331}{9851370771} a^{2} - \frac{829484965}{9851370771} a + \frac{1519062755}{3283790257}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{2267550436}{2393883097353} a^{15} - \frac{3100020956}{2393883097353} a^{14} + \frac{3253334549}{2393883097353} a^{13} - \frac{43870361789}{2393883097353} a^{12} + \frac{93470719823}{2393883097353} a^{11} - \frac{46341863384}{2393883097353} a^{10} + \frac{58490196379}{2393883097353} a^{9} - \frac{16621681918}{88662336939} a^{8} + \frac{5937568051}{24180637347} a^{7} - \frac{8680232483}{88662336939} a^{6} + \frac{46278696608}{88662336939} a^{5} - \frac{237812380427}{88662336939} a^{4} + \frac{498827983156}{88662336939} a^{3} - \frac{51925391549}{9851370771} a^{2} + \frac{42512515292}{9851370771} a - \frac{2494693813}{3283790257} \) (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:  \( 1003936.94223 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{37})\), 4.2.4107.1 x2, 4.0.333.1 x2, 8.0.151807041.1, 8.2.69274613043.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$37$37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$