↑

Properties

Label 16.0.33824788976...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{8}\cdot 101^{6}$
Root discriminant $45.51$
Ramified primes $5, 13, 101$
Class number $44$ (GRH)
Class group $[44]$ (GRH)
Galois group $C_2\wr C_2^2$ (as 16T128)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![824821, -2197228, 2937115, -2544195, 1554248, -673377, 186735, -12945, -16826, 9266, -2203, -29, 236, -109, 33, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 33*x^14 - 109*x^13 + 236*x^12 - 29*x^11 - 2203*x^10 + 9266*x^9 - 16826*x^8 - 12945*x^7 + 186735*x^6 - 673377*x^5 + 1554248*x^4 - 2544195*x^3 + 2937115*x^2 - 2197228*x + 824821)
 
gp: K = bnfinit(x^16 - 7*x^15 + 33*x^14 - 109*x^13 + 236*x^12 - 29*x^11 - 2203*x^10 + 9266*x^9 - 16826*x^8 - 12945*x^7 + 186735*x^6 - 673377*x^5 + 1554248*x^4 - 2544195*x^3 + 2937115*x^2 - 2197228*x + 824821, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 33 x^{14} - 109 x^{13} + 236 x^{12} - 29 x^{11} - 2203 x^{10} + 9266 x^{9} - 16826 x^{8} - 12945 x^{7} + 186735 x^{6} - 673377 x^{5} + 1554248 x^{4} - 2544195 x^{3} + 2937115 x^{2} - 2197228 x + 824821 \)

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:  \(338247889767883716141015625=5^{8}\cdot 13^{8}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.51$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{7}{16} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{5}{16}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} - \frac{7}{32} a^{8} + \frac{11}{64} a^{7} + \frac{19}{64} a^{6} - \frac{9}{64} a^{5} + \frac{1}{8} a^{4} - \frac{15}{64} a^{3} + \frac{11}{32} a^{2} + \frac{19}{64} a + \frac{27}{64}$, $\frac{1}{1280} a^{12} + \frac{1}{256} a^{11} + \frac{3}{256} a^{10} - \frac{67}{1280} a^{9} - \frac{279}{1280} a^{8} + \frac{7}{40} a^{7} + \frac{7}{320} a^{6} - \frac{631}{1280} a^{5} - \frac{11}{256} a^{4} - \frac{87}{256} a^{3} + \frac{73}{256} a^{2} + \frac{17}{40} a + \frac{541}{1280}$, $\frac{1}{5120} a^{13} - \frac{1}{512} a^{11} - \frac{71}{2560} a^{10} - \frac{73}{640} a^{9} + \frac{339}{5120} a^{8} + \frac{207}{1280} a^{7} + \frac{1149}{5120} a^{6} - \frac{69}{256} a^{5} - \frac{9}{32} a^{4} - \frac{1}{256} a^{3} + \frac{639}{5120} a^{2} - \frac{259}{5120} a - \frac{29}{1024}$, $\frac{1}{7542908067840} a^{14} - \frac{269216897}{7542908067840} a^{13} + \frac{337606881}{1257151344640} a^{12} + \frac{235485889}{628575672320} a^{11} + \frac{13828741247}{754290806784} a^{10} - \frac{107837646353}{1508581613568} a^{9} - \frac{141656935879}{7542908067840} a^{8} + \frac{1593724361537}{7542908067840} a^{7} + \frac{724074577427}{1508581613568} a^{6} - \frac{404781205019}{1885727016960} a^{5} - \frac{20431459783}{125715134464} a^{4} + \frac{229194929059}{7542908067840} a^{3} - \frac{1062988825081}{3771454033920} a^{2} - \frac{321512518573}{1257151344640} a - \frac{2279566097119}{7542908067840}$, $\frac{1}{1840469568552960} a^{15} - \frac{53}{920234784276480} a^{14} - \frac{89708561969}{1840469568552960} a^{13} - \frac{105964461831}{306744928092160} a^{12} - \frac{5865920035931}{920234784276480} a^{11} + \frac{50412833868421}{1840469568552960} a^{10} + \frac{31300446158219}{920234784276480} a^{9} + \frac{4307106360289}{28757337008640} a^{8} - \frac{13554486953527}{306744928092160} a^{7} + \frac{579125135015357}{1840469568552960} a^{6} - \frac{3126835873387}{230058696069120} a^{5} - \frac{865750009296281}{1840469568552960} a^{4} + \frac{24243875186867}{1840469568552960} a^{3} + \frac{140560193034277}{460117392138240} a^{2} - \frac{89483565626189}{368093913710592} a - \frac{46264695905449}{1840469568552960}$

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

Class group and class number

$C_{44}$, which has order $44$ (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:  \( 229349.978606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T128):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), 4.4.426725.2, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.426725.4, 8.0.1802913125.1, 8.0.18391516788125.3, 8.8.182094225625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$