↑

Properties

Label 16.0.70744959183...8256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 7^{8}\cdot 17^{8}$
Root discriminant $73.38$
Ramified primes $2, 7, 17$
Class number $1560$ (GRH)
Class group $[2, 2, 390]$ (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, 0, 48020, 0, 165718, 0, 189924, 0, 83671, 0, 16268, 0, 1502, 0, 64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 64*x^14 + 1502*x^12 + 16268*x^10 + 83671*x^8 + 189924*x^6 + 165718*x^4 + 48020*x^2 + 2401)
 
gp: K = bnfinit(x^16 + 64*x^14 + 1502*x^12 + 16268*x^10 + 83671*x^8 + 189924*x^6 + 165718*x^4 + 48020*x^2 + 2401, 1)
 

Normalized defining polynomial

\( x^{16} + 64 x^{14} + 1502 x^{12} + 16268 x^{10} + 83671 x^{8} + 189924 x^{6} + 165718 x^{4} + 48020 x^{2} + 2401 \)

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:  \(707449591835873807436932448256=2^{44}\cdot 7^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.38$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{8} + \frac{1}{14} a^{6} - \frac{1}{2} a^{5} - \frac{3}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{9} + \frac{1}{14} a^{7} + \frac{2}{7} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{98} a^{10} + \frac{1}{98} a^{8} - \frac{5}{49} a^{6} - \frac{1}{2} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{3} - \frac{5}{14} a^{2} - \frac{1}{2} a$, $\frac{1}{98} a^{11} + \frac{1}{98} a^{9} - \frac{5}{49} a^{7} + \frac{1}{7} a^{5} - \frac{1}{2} a^{4} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{196} a^{12} - \frac{1}{196} a^{10} + \frac{1}{98} a^{8} - \frac{1}{196} a^{6} + \frac{3}{14} a^{4} - \frac{11}{28} a^{2} + \frac{1}{4}$, $\frac{1}{196} a^{13} - \frac{1}{196} a^{11} + \frac{1}{98} a^{9} - \frac{1}{196} a^{7} + \frac{3}{14} a^{5} - \frac{11}{28} a^{3} + \frac{1}{4} a$, $\frac{1}{1573124653436} a^{14} - \frac{634035512}{393281163359} a^{12} - \frac{5545093235}{1573124653436} a^{10} + \frac{1765066523}{224732093348} a^{8} + \frac{5990066025}{224732093348} a^{6} - \frac{1}{2} a^{5} - \frac{15655739847}{32104584764} a^{4} - \frac{1}{2} a^{3} + \frac{1757105732}{8026146191} a^{2} - \frac{1}{2} a + \frac{603094349}{4586369252}$, $\frac{1}{1573124653436} a^{15} - \frac{634035512}{393281163359} a^{13} - \frac{5545093235}{1573124653436} a^{11} + \frac{1765066523}{224732093348} a^{9} + \frac{5990066025}{224732093348} a^{7} + \frac{396552535}{32104584764} a^{5} - \frac{1}{2} a^{4} - \frac{4511934727}{16052292382} a^{3} - \frac{1}{2} a^{2} - \frac{1690090277}{4586369252} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{390}$, which has order $1560$ (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:  \( 202775.215369 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.32368.1, 8.8.268207980544.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$