↑

Properties

Label 16.0.59488299467...2809.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{11}$
Root discriminant $72.59$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11461, 30868, 57639, 27680, -16769, -39502, 4628, 16546, 5408, -6950, 472, 550, 95, -94, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 94*x^13 + 95*x^12 + 550*x^11 + 472*x^10 - 6950*x^9 + 5408*x^8 + 16546*x^7 + 4628*x^6 - 39502*x^5 - 16769*x^4 + 27680*x^3 + 57639*x^2 + 30868*x + 11461)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 94*x^13 + 95*x^12 + 550*x^11 + 472*x^10 - 6950*x^9 + 5408*x^8 + 16546*x^7 + 4628*x^6 - 39502*x^5 - 16769*x^4 + 27680*x^3 + 57639*x^2 + 30868*x + 11461, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 94 x^{13} + 95 x^{12} + 550 x^{11} + 472 x^{10} - 6950 x^{9} + 5408 x^{8} + 16546 x^{7} + 4628 x^{6} - 39502 x^{5} - 16769 x^{4} + 27680 x^{3} + 57639 x^{2} + 30868 x + 11461 \)

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:  \(594882994674022504125031032809=11^{8}\cdot 89^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.59$
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:  $11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{32} a^{9} - \frac{3}{32} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{7}{32} a^{3} - \frac{1}{8} a - \frac{13}{32}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{7}{32} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{15}{32} a + \frac{1}{8}$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{8} - \frac{1}{32} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{9}{32} a^{2} - \frac{1}{8}$, $\frac{1}{256} a^{12} + \frac{1}{128} a^{10} - \frac{1}{64} a^{9} + \frac{1}{32} a^{8} - \frac{5}{128} a^{7} + \frac{13}{128} a^{6} - \frac{1}{8} a^{5} - \frac{9}{128} a^{4} + \frac{11}{64} a^{3} + \frac{9}{32} a^{2} - \frac{27}{128} a + \frac{45}{256}$, $\frac{1}{256} a^{13} + \frac{1}{128} a^{11} - \frac{1}{64} a^{10} - \frac{5}{128} a^{8} - \frac{3}{128} a^{7} + \frac{3}{32} a^{6} - \frac{9}{128} a^{5} - \frac{13}{64} a^{4} - \frac{3}{16} a^{3} + \frac{5}{128} a^{2} - \frac{19}{256} a - \frac{15}{32}$, $\frac{1}{135168} a^{14} + \frac{1}{45056} a^{13} - \frac{29}{45056} a^{12} - \frac{443}{67584} a^{11} + \frac{299}{22528} a^{10} + \frac{151}{22528} a^{9} - \frac{377}{33792} a^{8} - \frac{43}{704} a^{7} + \frac{1053}{11264} a^{6} - \frac{7465}{67584} a^{5} - \frac{15973}{67584} a^{4} + \frac{9391}{67584} a^{3} + \frac{1769}{45056} a^{2} + \frac{53653}{135168} a - \frac{22621}{135168}$, $\frac{1}{430831424473864753152} a^{15} - \frac{20254245258595}{13463482014808273536} a^{14} - \frac{2234140757788299}{4487827338269424512} a^{13} - \frac{480019652797560625}{430831424473864753152} a^{12} - \frac{189463964462993575}{107707856118466188288} a^{11} + \frac{458293703630276203}{35902618706155396096} a^{10} - \frac{2534703965005025185}{215415712236932376576} a^{9} + \frac{4504650457703556427}{107707856118466188288} a^{8} + \frac{72402213434322459}{1560983422006756352} a^{7} - \frac{18786039508578450451}{215415712236932376576} a^{6} - \frac{24727045663602102709}{107707856118466188288} a^{5} - \frac{454507789502382267}{35902618706155396096} a^{4} - \frac{33240356807508373663}{430831424473864753152} a^{3} + \frac{52534761276614865601}{107707856118466188288} a^{2} - \frac{2899122350121080669}{35902618706155396096} a + \frac{192544191234023565719}{430831424473864753152}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2250798584.84 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.2.87131.1, 8.0.675671193329.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.3.2$x^{4} - 801$$4$$1$$3$$C_4$$[\ ]_{4}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$