↑

Properties

Label 16.0.33260226740...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{6}\cdot 11^{2}$
Root discriminant $16.60$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group 16T1220

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 96, 432, 960, 1192, 592, -240, -480, -240, 16, 72, 16, -4, 8, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^13 - 4*x^12 + 16*x^11 + 72*x^10 + 16*x^9 - 240*x^8 - 480*x^7 - 240*x^6 + 592*x^5 + 1192*x^4 + 960*x^3 + 432*x^2 + 96*x + 8)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^13 - 4*x^12 + 16*x^11 + 72*x^10 + 16*x^9 - 240*x^8 - 480*x^7 - 240*x^6 + 592*x^5 + 1192*x^4 + 960*x^3 + 432*x^2 + 96*x + 8, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{13} - 4 x^{12} + 16 x^{11} + 72 x^{10} + 16 x^{9} - 240 x^{8} - 480 x^{7} - 240 x^{6} + 592 x^{5} + 1192 x^{4} + 960 x^{3} + 432 x^{2} + 96 x + 8 \)

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:  \(33260226740224000000=2^{44}\cdot 5^{6}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.60$
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 not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{20} a^{11} + \frac{1}{5} a^{8} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{20} a^{12} + \frac{1}{5} a^{9} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{220} a^{14} + \frac{1}{55} a^{13} + \frac{1}{44} a^{12} - \frac{1}{44} a^{11} + \frac{2}{55} a^{10} + \frac{7}{55} a^{9} - \frac{3}{22} a^{8} - \frac{3}{55} a^{7} - \frac{27}{110} a^{6} + \frac{7}{55} a^{5} - \frac{19}{55} a^{4} + \frac{27}{55} a^{3} + \frac{26}{55} a^{2} - \frac{17}{55} a - \frac{4}{55}$, $\frac{1}{7276223900} a^{15} + \frac{14219977}{7276223900} a^{14} - \frac{46960323}{7276223900} a^{13} + \frac{301}{118844} a^{12} + \frac{71253651}{7276223900} a^{11} + \frac{10257778}{1819055975} a^{10} + \frac{138393756}{1819055975} a^{9} + \frac{24577869}{145524478} a^{8} - \frac{43839176}{363811195} a^{7} - \frac{101231629}{727622390} a^{6} + \frac{15783504}{33073745} a^{5} - \frac{495121562}{1819055975} a^{4} + \frac{896525426}{1819055975} a^{3} - \frac{895338689}{1819055975} a^{2} - \frac{663815056}{1819055975} a + \frac{571559678}{1819055975}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{58159053}{99674300} a^{15} - \frac{123238717}{49837150} a^{14} + \frac{28903843}{49837150} a^{13} + \frac{37201}{8140} a^{12} - \frac{171824691}{49837150} a^{11} + \frac{251595844}{24918575} a^{10} + \frac{989018533}{24918575} a^{9} - \frac{1572331}{4983715} a^{8} - \frac{699675348}{4983715} a^{7} - \frac{245685079}{996743} a^{6} - \frac{395121043}{4983715} a^{5} + \frac{9148118124}{24918575} a^{4} + \frac{15146450688}{24918575} a^{3} + \frac{10202772703}{24918575} a^{2} + \frac{3678141582}{24918575} a + \frac{40904679}{2265325} \) (order $8$)
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:  \( 6998.47908429 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.1280.1, 4.0.320.1, \(\Q(\zeta_{8})\), 8.0.6553600.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$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}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$