↑

Properties

Label 16.0.22825217147...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $51.27$
Ramified primes $2, 3, 5$
Class number $320$ (GRH)
Class group $[2, 2, 80]$ (GRH)
Galois group $C_4.D_4$ (as 16T30)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167101, -199044, 75320, -54488, 136322, -142404, 82820, -30156, 7441, -2436, 1516, -652, 156, -16, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 16*x^13 + 156*x^12 - 652*x^11 + 1516*x^10 - 2436*x^9 + 7441*x^8 - 30156*x^7 + 82820*x^6 - 142404*x^5 + 136322*x^4 - 54488*x^3 + 75320*x^2 - 199044*x + 167101)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 16*x^13 + 156*x^12 - 652*x^11 + 1516*x^10 - 2436*x^9 + 7441*x^8 - 30156*x^7 + 82820*x^6 - 142404*x^5 + 136322*x^4 - 54488*x^3 + 75320*x^2 - 199044*x + 167101, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 16 x^{13} + 156 x^{12} - 652 x^{11} + 1516 x^{10} - 2436 x^{9} + 7441 x^{8} - 30156 x^{7} + 82820 x^{6} - 142404 x^{5} + 136322 x^{4} - 54488 x^{3} + 75320 x^{2} - 199044 x + 167101 \)

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:  \(2282521714753536000000000000=2^{44}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.27$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{21} a^{12} + \frac{1}{21} a^{10} - \frac{2}{21} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{3} a^{6} + \frac{2}{7} a^{5} - \frac{8}{21} a^{4} - \frac{4}{21} a^{3} - \frac{3}{7} a^{2} + \frac{4}{21} a + \frac{1}{3}$, $\frac{1}{21} a^{13} + \frac{1}{21} a^{11} - \frac{2}{21} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{3} a^{7} + \frac{2}{7} a^{6} - \frac{8}{21} a^{5} - \frac{4}{21} a^{4} - \frac{3}{7} a^{3} + \frac{4}{21} a^{2} + \frac{1}{3} a$, $\frac{1}{21} a^{14} + \frac{1}{21} a^{11} + \frac{5}{21} a^{10} + \frac{8}{21} a^{9} - \frac{5}{21} a^{8} - \frac{2}{7} a^{6} - \frac{10}{21} a^{5} - \frac{4}{21} a^{4} + \frac{5}{21} a^{3} + \frac{4}{21} a^{2} - \frac{10}{21} a + \frac{5}{21}$, $\frac{1}{8686837821747458349236032519467} a^{15} - \frac{25314057898536775839672783935}{2895612607249152783078677506489} a^{14} - \frac{1238709652176560926335992909}{59094134841819444552626071561} a^{13} + \frac{128874457864020460496295578773}{8686837821747458349236032519467} a^{12} + \frac{335717721432441244973606743652}{8686837821747458349236032519467} a^{11} - \frac{2779100355464286986886546523834}{8686837821747458349236032519467} a^{10} + \frac{2334414338514737323731569258290}{8686837821747458349236032519467} a^{9} - \frac{500480379692664429425811972175}{2895612607249152783078677506489} a^{8} + \frac{128404107548989350231975461380}{2895612607249152783078677506489} a^{7} + \frac{3474627554640613824181243320086}{8686837821747458349236032519467} a^{6} + \frac{2856767109056001228493506720107}{8686837821747458349236032519467} a^{5} + \frac{118546241437570844775794925809}{1240976831678208335605147502781} a^{4} + \frac{872103219849772250638595821885}{8686837821747458349236032519467} a^{3} - \frac{1821928570386752764893202150288}{8686837821747458349236032519467} a^{2} - \frac{3757832417374562060300787781948}{8686837821747458349236032519467} a + \frac{525120114550414681634574736691}{2895612607249152783078677506489}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{80}$, which has order $320$ (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:  \( 55877.3346526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

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 $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), 4.0.576000.1, \(\Q(\sqrt{3}, \sqrt{5})\), 4.0.576000.2, 8.0.331776000000.26, 8.8.29859840000.1, 8.0.2985984000000.1

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 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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
5Data not computed