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Properties

Label 16.0.15772350630...5936.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 23^{4}\cdot 107^{4}$
Root discriminant $24.40$
Ramified primes $2, 3, 23, 107$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^2:S_4:C_2$ (as 16T724)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 672, 1320, 1524, 1302, 884, 520, 120, 58, -12, 16, -38, 46, -32, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 32*x^13 + 46*x^12 - 38*x^11 + 16*x^10 - 12*x^9 + 58*x^8 + 120*x^7 + 520*x^6 + 884*x^5 + 1302*x^4 + 1524*x^3 + 1320*x^2 + 672*x + 169)
 
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 32*x^13 + 46*x^12 - 38*x^11 + 16*x^10 - 12*x^9 + 58*x^8 + 120*x^7 + 520*x^6 + 884*x^5 + 1302*x^4 + 1524*x^3 + 1320*x^2 + 672*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 17 x^{14} - 32 x^{13} + 46 x^{12} - 38 x^{11} + 16 x^{10} - 12 x^{9} + 58 x^{8} + 120 x^{7} + 520 x^{6} + 884 x^{5} + 1302 x^{4} + 1524 x^{3} + 1320 x^{2} + 672 x + 169 \)

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:  \(15772350630880634535936=2^{16}\cdot 3^{8}\cdot 23^{4}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.40$
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:  $2, 3, 23, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{8} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{2}{27} a^{9} - \frac{2}{27} a^{8} + \frac{1}{27} a^{7} + \frac{13}{27} a^{6} + \frac{13}{27} a^{5} - \frac{1}{27} a^{4} - \frac{8}{27} a^{3} - \frac{2}{9} a^{2} - \frac{5}{27} a - \frac{7}{27}$, $\frac{1}{81} a^{12} + \frac{1}{81} a^{11} + \frac{2}{81} a^{9} - \frac{1}{27} a^{8} + \frac{5}{27} a^{7} - \frac{5}{27} a^{6} - \frac{2}{81} a^{5} + \frac{17}{81} a^{4} + \frac{32}{81} a^{3} + \frac{10}{81} a^{2} + \frac{37}{81} a - \frac{14}{81}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} + \frac{2}{243} a^{10} - \frac{5}{243} a^{9} + \frac{2}{27} a^{8} + \frac{17}{81} a^{7} + \frac{94}{243} a^{6} - \frac{62}{243} a^{5} + \frac{32}{81} a^{4} - \frac{103}{243} a^{3} + \frac{4}{9} a^{2} - \frac{17}{81} a + \frac{95}{243}$, $\frac{1}{8019} a^{14} - \frac{4}{8019} a^{13} + \frac{4}{729} a^{12} - \frac{28}{2673} a^{11} - \frac{364}{8019} a^{10} - \frac{142}{8019} a^{9} - \frac{772}{2673} a^{8} - \frac{226}{729} a^{7} - \frac{1037}{2673} a^{6} - \frac{2230}{8019} a^{5} - \frac{559}{8019} a^{4} + \frac{2797}{8019} a^{3} - \frac{65}{2673} a^{2} - \frac{763}{8019} a - \frac{794}{8019}$, $\frac{1}{24057} a^{15} + \frac{1}{24057} a^{14} + \frac{8}{8019} a^{13} + \frac{136}{24057} a^{12} + \frac{107}{24057} a^{11} - \frac{20}{2673} a^{10} - \frac{1244}{24057} a^{9} - \frac{10502}{24057} a^{8} + \frac{1388}{24057} a^{7} - \frac{3529}{24057} a^{6} - \frac{1202}{2673} a^{5} - \frac{889}{24057} a^{4} + \frac{3989}{24057} a^{3} + \frac{85}{2187} a^{2} - \frac{95}{2187} a + \frac{8504}{24057}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{5}{27} a^{15} - \frac{9445}{8019} a^{14} + \frac{28210}{8019} a^{13} - \frac{4948}{729} a^{12} + \frac{25397}{2673} a^{11} - \frac{56060}{8019} a^{10} - \frac{5078}{8019} a^{9} + \frac{15238}{2673} a^{8} + \frac{1453}{729} a^{7} + \frac{68419}{2673} a^{6} + \frac{664027}{8019} a^{5} + \frac{1051564}{8019} a^{4} + \frac{1387739}{8019} a^{3} + \frac{541259}{2673} a^{2} + \frac{1117540}{8019} a + \frac{362090}{8019} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 89581.368292 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:S_4:C_2$ (as 16T724):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$
Character table for $C_2\times C_2^2:S_4:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.8.125588019456.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$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}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$