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Properties

Label 16.0.10044752324...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 5^{4}\cdot 7^{6}\cdot 13^{4}$
Root discriminant $15.40$
Ramified primes $3, 5, 7, 13$
Class number $2$
Class group $[2]$
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -9, -9, -63, 162, -117, 135, -234, 159, -84, 99, -54, 15, -15, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 15*x^13 + 15*x^12 - 54*x^11 + 99*x^10 - 84*x^9 + 159*x^8 - 234*x^7 + 135*x^6 - 117*x^5 + 162*x^4 - 63*x^3 - 9*x^2 - 9*x + 9)
 
gp: K = bnfinit(x^16 + 6*x^14 - 15*x^13 + 15*x^12 - 54*x^11 + 99*x^10 - 84*x^9 + 159*x^8 - 234*x^7 + 135*x^6 - 117*x^5 + 162*x^4 - 63*x^3 - 9*x^2 - 9*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 15 x^{13} + 15 x^{12} - 54 x^{11} + 99 x^{10} - 84 x^{9} + 159 x^{8} - 234 x^{7} + 135 x^{6} - 117 x^{5} + 162 x^{4} - 63 x^{3} - 9 x^{2} - 9 x + 9 \)

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:  \(10044752324575775625=3^{14}\cdot 5^{4}\cdot 7^{6}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.40$
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:  $3, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{3} a^{13}$, $\frac{1}{39} a^{14} - \frac{5}{39} a^{13} + \frac{2}{39} a^{12} + \frac{1}{13} a^{11} - \frac{2}{13} a^{10} + \frac{2}{13} a^{9} - \frac{4}{39} a^{8} + \frac{3}{13} a^{7} + \frac{3}{13} a^{6} + \frac{2}{13} a^{5} - \frac{6}{13} a^{3} + \frac{6}{13} a^{2} + \frac{6}{13} a + \frac{1}{13}$, $\frac{1}{1303497} a^{15} + \frac{2272}{434499} a^{14} - \frac{11177}{434499} a^{13} - \frac{93}{144833} a^{12} - \frac{18830}{434499} a^{11} - \frac{23260}{434499} a^{10} + \frac{17336}{144833} a^{9} - \frac{64684}{434499} a^{8} - \frac{159272}{434499} a^{7} + \frac{71657}{144833} a^{6} + \frac{37251}{144833} a^{5} + \frac{10554}{144833} a^{4} - \frac{45919}{144833} a^{3} + \frac{202}{144833} a^{2} - \frac{71499}{144833} a + \frac{25860}{144833}$

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:  \( \frac{86551}{33423} a^{15} + \frac{67943}{33423} a^{14} + \frac{191327}{11141} a^{13} - \frac{846836}{33423} a^{12} + \frac{214106}{11141} a^{11} - \frac{1394470}{11141} a^{10} + \frac{1766177}{11141} a^{9} - \frac{81424}{857} a^{8} + \frac{291320}{857} a^{7} - \frac{3803407}{11141} a^{6} + \frac{969780}{11141} a^{5} - \frac{2681271}{11141} a^{4} + \frac{2608245}{11141} a^{3} + \frac{14553}{857} a^{2} - \frac{78219}{11141} a - \frac{324932}{11141} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1130.60888908 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.2, 8.0.18753525.1, 8.0.3169345725.3, 8.0.6036849.1

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

Sibling fields

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 R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$