↑

Properties

Label 16.0.23125079889...0501.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{11}$
Root discriminant $38.48$
Ramified primes $3, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 369, 1053, 3192, 2052, 4701, 375, 2887, -690, 1004, -375, 255, -84, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 32*x^14 - 84*x^13 + 255*x^12 - 375*x^11 + 1004*x^10 - 690*x^9 + 2887*x^8 + 375*x^7 + 4701*x^6 + 2052*x^5 + 3192*x^4 + 1053*x^3 + 369*x^2 + 9)
 
gp: K = bnfinit(x^16 - 6*x^15 + 32*x^14 - 84*x^13 + 255*x^12 - 375*x^11 + 1004*x^10 - 690*x^9 + 2887*x^8 + 375*x^7 + 4701*x^6 + 2052*x^5 + 3192*x^4 + 1053*x^3 + 369*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 32 x^{14} - 84 x^{13} + 255 x^{12} - 375 x^{11} + 1004 x^{10} - 690 x^{9} + 2887 x^{8} + 375 x^{7} + 4701 x^{6} + 2052 x^{5} + 3192 x^{4} + 1053 x^{3} + 369 x^{2} + 9 \)

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:  \(23125079889339073533840501=3^{12}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.48$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{6759} a^{14} - \frac{1042}{6759} a^{13} - \frac{220}{2253} a^{12} - \frac{48}{751} a^{11} - \frac{238}{2253} a^{10} + \frac{869}{2253} a^{9} + \frac{2171}{6759} a^{8} - \frac{890}{6759} a^{7} - \frac{341}{2253} a^{6} - \frac{512}{2253} a^{5} - \frac{625}{2253} a^{4} - \frac{419}{2253} a^{3} - \frac{51}{751} a^{2} - \frac{266}{751} a + \frac{48}{751}$, $\frac{1}{74702705468369985} a^{15} + \frac{1156430578703}{74702705468369985} a^{14} - \frac{3178363650059507}{24900901822789995} a^{13} + \frac{843942016231183}{8300300607596665} a^{12} + \frac{5999222792108}{76149546858685} a^{11} + \frac{779487381424718}{8300300607596665} a^{10} + \frac{2009908414653812}{74702705468369985} a^{9} + \frac{24399819829248958}{74702705468369985} a^{8} + \frac{2541003848376913}{24900901822789995} a^{7} + \frac{5639735251002992}{24900901822789995} a^{6} - \frac{273459836966683}{1660060121519333} a^{5} + \frac{5798142138712474}{24900901822789995} a^{4} + \frac{758904870332696}{1660060121519333} a^{3} - \frac{1029282779270188}{8300300607596665} a^{2} + \frac{2870712352845009}{8300300607596665} a + \frac{1633215681478596}{8300300607596665}$

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:  \( 1926849.39647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.2.11163.1, 8.0.68412300381.1

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 $16$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ $16$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
61Data not computed