↑

Properties

Label 16.0.33708728269...9277.4
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 37^{9}$
Root discriminant $34.12$
Ramified primes $11, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -1050, 1815, -1692, 850, 110, -901, 945, -845, 442, 3, -166, 191, -79, 31, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 31*x^14 - 79*x^13 + 191*x^12 - 166*x^11 + 3*x^10 + 442*x^9 - 845*x^8 + 945*x^7 - 901*x^6 + 110*x^5 + 850*x^4 - 1692*x^3 + 1815*x^2 - 1050*x + 625)
 
gp: K = bnfinit(x^16 - 5*x^15 + 31*x^14 - 79*x^13 + 191*x^12 - 166*x^11 + 3*x^10 + 442*x^9 - 845*x^8 + 945*x^7 - 901*x^6 + 110*x^5 + 850*x^4 - 1692*x^3 + 1815*x^2 - 1050*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 31 x^{14} - 79 x^{13} + 191 x^{12} - 166 x^{11} + 3 x^{10} + 442 x^{9} - 845 x^{8} + 945 x^{7} - 901 x^{6} + 110 x^{5} + 850 x^{4} - 1692 x^{3} + 1815 x^{2} - 1050 x + 625 \)

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:  \(3370872826949590878489277=11^{10}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.12$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} - \frac{2}{11} a^{7} + \frac{1}{11} a^{6} + \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{55} a^{13} + \frac{1}{55} a^{12} + \frac{2}{55} a^{11} - \frac{17}{55} a^{10} - \frac{21}{55} a^{9} + \frac{13}{55} a^{8} - \frac{24}{55} a^{7} + \frac{18}{55} a^{6} + \frac{13}{55} a^{5} - \frac{2}{5} a^{4} + \frac{27}{55} a^{3} + \frac{2}{55} a^{2} - \frac{3}{55} a + \frac{2}{11}$, $\frac{1}{2035} a^{14} + \frac{16}{2035} a^{13} + \frac{7}{2035} a^{12} - \frac{697}{2035} a^{11} + \frac{54}{2035} a^{10} + \frac{43}{185} a^{9} - \frac{809}{2035} a^{8} + \frac{393}{2035} a^{7} - \frac{497}{2035} a^{6} + \frac{173}{2035} a^{5} + \frac{722}{2035} a^{4} - \frac{388}{2035} a^{3} - \frac{1003}{2035} a^{2} + \frac{1}{407} a - \frac{31}{407}$, $\frac{1}{4880218843648854475} a^{15} + \frac{209034378665747}{976043768729770895} a^{14} + \frac{3017582537638071}{4880218843648854475} a^{13} + \frac{20743668545324}{7519597601924275} a^{12} - \frac{319041416635994152}{697174120521264925} a^{11} - \frac{1460823654891420531}{4880218843648854475} a^{10} - \frac{704138904994927}{1573756479731975} a^{9} + \frac{1906783385794953807}{4880218843648854475} a^{8} - \frac{71031108286782860}{195208753745954179} a^{7} - \frac{36403121050624462}{88731251702706445} a^{6} + \frac{1069131233039671519}{4880218843648854475} a^{5} + \frac{13605996424584733}{88731251702706445} a^{4} - \frac{213403551342112114}{976043768729770895} a^{3} - \frac{1040411891307385187}{4880218843648854475} a^{2} - \frac{480801589326193147}{976043768729770895} a - \frac{55711036371519167}{195208753745954179}$

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:  \( 326494.750386 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.4477.1, 8.0.741610573.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed