↑

Properties

Label 16.0.48696297970...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{10}\cdot 11^{6}$
Root discriminant $53.76$
Ramified primes $2, 5, 11$
Class number $932$ (GRH)
Class group $[2, 466]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T293)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48400, 0, 193600, 0, 265760, 0, 174240, 0, 60824, 0, 11568, 0, 1160, 0, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 56*x^14 + 1160*x^12 + 11568*x^10 + 60824*x^8 + 174240*x^6 + 265760*x^4 + 193600*x^2 + 48400)
 
gp: K = bnfinit(x^16 + 56*x^14 + 1160*x^12 + 11568*x^10 + 60824*x^8 + 174240*x^6 + 265760*x^4 + 193600*x^2 + 48400, 1)
 

Normalized defining polynomial

\( x^{16} + 56 x^{14} + 1160 x^{12} + 11568 x^{10} + 60824 x^{8} + 174240 x^{6} + 265760 x^{4} + 193600 x^{2} + 48400 \)

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:  \(4869629797036195840000000000=2^{48}\cdot 5^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.76$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{32} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{32} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{32} a^{10} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{11} - \frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{7}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{7040} a^{12} + \frac{7}{880} a^{10} - \frac{5}{704} a^{8} + \frac{1}{55} a^{6} + \frac{191}{1760} a^{4} - \frac{1}{4} a^{2} - \frac{1}{16}$, $\frac{1}{14080} a^{13} + \frac{7}{1760} a^{11} + \frac{17}{1408} a^{9} + \frac{1}{110} a^{7} - \frac{29}{3520} a^{5} + \frac{3}{8} a^{3} - \frac{15}{32} a$, $\frac{1}{11080960} a^{14} + \frac{41}{1108096} a^{12} + \frac{26717}{5540480} a^{10} - \frac{31123}{2770240} a^{8} - \frac{59101}{2770240} a^{6} - \frac{151433}{1385120} a^{4} - \frac{5191}{25184} a^{2} - \frac{2519}{12592}$, $\frac{1}{11080960} a^{15} - \frac{377}{11080960} a^{13} + \frac{4681}{5540480} a^{11} + \frac{43999}{5540480} a^{9} - \frac{16857}{554048} a^{7} + \frac{66237}{2770240} a^{5} + \frac{10549}{25184} a^{3} - \frac{2677}{25184} a$

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

Class group and class number

$C_{2}\times C_{466}$, which has order $932$ (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:  \( 17367.3059334 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T293):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.17600.1, 4.4.4400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.4956160000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$