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Properties

Label 16.0.42445765882...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 19^{4}\cdot 89^{6}$
Root discriminant $71.08$
Ramified primes $2, 5, 19, 89$
Class number $1056$ (GRH)
Class group $[2, 2, 264]$ (GRH)
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![167281, 0, -62647, 0, 188752, 0, -28047, 0, 21398, 0, 219, 0, 328, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 + 328*x^12 + 219*x^10 + 21398*x^8 - 28047*x^6 + 188752*x^4 - 62647*x^2 + 167281)
 
gp: K = bnfinit(x^16 + 11*x^14 + 328*x^12 + 219*x^10 + 21398*x^8 - 28047*x^6 + 188752*x^4 - 62647*x^2 + 167281, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} + 328 x^{12} + 219 x^{10} + 21398 x^{8} - 28047 x^{6} + 188752 x^{4} - 62647 x^{2} + 167281 \)

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:  \(424457658822351133081600000000=2^{24}\cdot 5^{8}\cdot 19^{4}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.08$
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:  $2, 5, 19, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{3}{8}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{7} + \frac{3}{16} a^{6} - \frac{5}{16} a^{5} + \frac{5}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{16} a - \frac{1}{16}$, $\frac{1}{352} a^{12} + \frac{13}{352} a^{10} + \frac{37}{352} a^{8} + \frac{1}{8} a^{6} - \frac{9}{32} a^{4} + \frac{47}{352} a^{2} - \frac{123}{352}$, $\frac{1}{352} a^{13} - \frac{9}{352} a^{11} - \frac{1}{16} a^{10} + \frac{37}{352} a^{9} - \frac{3}{16} a^{7} + \frac{3}{16} a^{6} + \frac{1}{32} a^{5} + \frac{5}{16} a^{4} + \frac{91}{352} a^{3} + \frac{1}{8} a^{2} + \frac{31}{352} a - \frac{1}{16}$, $\frac{1}{29545674780774592} a^{14} - \frac{3954945608409}{3693209347596824} a^{12} + \frac{54413433004321}{7386418695193648} a^{10} - \frac{294242771520317}{29545674780774592} a^{8} - \frac{16151788741521}{92619670159168} a^{6} - \frac{6041091824874561}{14772837390387296} a^{4} + \frac{5291891767726677}{14772837390387296} a^{2} + \frac{11372408453266951}{29545674780774592}$, $\frac{1}{24168361970673616256} a^{15} - \frac{1}{59091349561549184} a^{14} + \frac{6069581983492949}{12084180985336808128} a^{13} - \frac{26148505607237}{29545674780774592} a^{12} + \frac{58906398411271715}{12084180985336808128} a^{11} + \frac{1192190063258421}{29545674780774592} a^{10} - \frac{288616381612317827}{24168361970673616256} a^{9} - \frac{2811410543504285}{59091349561549184} a^{8} + \frac{562721149753279}{75762890190199424} a^{7} - \frac{30158046338063}{185239340318336} a^{6} + \frac{623570362453922537}{1510522623167101016} a^{5} + \frac{30091530372779}{923302336899206} a^{4} - \frac{242019653191060203}{3021045246334202032} a^{3} + \frac{953806684285691}{7386418695193648} a^{2} - \frac{48977989749508423}{24168361970673616256} a + \frac{2644999752384631}{59091349561549184}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{264}$, which has order $1056$ (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:  \( 245070.279863 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.4.2225.1, 4.4.7600.1, 4.4.676400.1, 8.0.10179752360000.1, 8.0.451180160000.1, 8.8.457516960000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$