↑

Properties

Label 16.0.19244198650...8721.5
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 41^{12}$
Root discriminant $90.21$
Ramified primes $31, 41$
Class number $240$ (GRH)
Class group $[2, 2, 60]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![600625, 0, 3204625, 0, 2258879, 0, 1029206, 0, 80135, 0, 3210, 0, 570, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 44*x^14 + 570*x^12 + 3210*x^10 + 80135*x^8 + 1029206*x^6 + 2258879*x^4 + 3204625*x^2 + 600625)
 
gp: K = bnfinit(x^16 + 44*x^14 + 570*x^12 + 3210*x^10 + 80135*x^8 + 1029206*x^6 + 2258879*x^4 + 3204625*x^2 + 600625, 1)
 

Normalized defining polynomial

\( x^{16} + 44 x^{14} + 570 x^{12} + 3210 x^{10} + 80135 x^{8} + 1029206 x^{6} + 2258879 x^{4} + 3204625 x^{2} + 600625 \)

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:  \(19244198650569257148820544058721=31^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.21$
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:  $31, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{50} a^{8} - \frac{2}{25} a^{6} + \frac{19}{50} a^{4} - \frac{1}{50} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{50} a^{9} - \frac{2}{25} a^{7} - \frac{1}{50} a^{5} - \frac{1}{50} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a$, $\frac{1}{50} a^{10} + \frac{3}{50} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{21}{50} a^{2}$, $\frac{1}{250} a^{11} + \frac{1}{125} a^{9} - \frac{1}{50} a^{7} + \frac{13}{250} a^{5} - \frac{1}{2} a^{4} + \frac{69}{250} a^{3} + \frac{1}{5} a$, $\frac{1}{1250} a^{12} + \frac{1}{625} a^{10} - \frac{1}{250} a^{8} - \frac{87}{1250} a^{6} - \frac{1}{10} a^{5} + \frac{569}{1250} a^{4} - \frac{12}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{1250} a^{13} + \frac{1}{625} a^{11} - \frac{1}{250} a^{9} - \frac{87}{1250} a^{7} - \frac{1}{10} a^{6} + \frac{69}{1250} a^{5} - \frac{12}{25} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{1760099123915743750} a^{14} + \frac{68551133888233}{1760099123915743750} a^{12} + \frac{10964689052109607}{1760099123915743750} a^{10} - \frac{3587669450752317}{1760099123915743750} a^{8} - \frac{1}{10} a^{7} - \frac{1903158962778819}{28388695547028125} a^{6} - \frac{1}{10} a^{5} - \frac{58551866922511918}{880049561957871875} a^{4} - \frac{2}{5} a^{3} - \frac{3414431643328181}{14080792991325950} a^{2} + \frac{1}{10} a + \frac{7458268704211}{90843825750490}$, $\frac{1}{1760099123915743750} a^{15} + \frac{68551133888233}{1760099123915743750} a^{13} - \frac{3116103939216343}{1760099123915743750} a^{11} + \frac{1726363522455329}{880049561957871875} a^{9} + \frac{2639032324745681}{28388695547028125} a^{7} - \frac{1}{10} a^{6} + \frac{16663799572572689}{1760099123915743750} a^{5} - \frac{1}{2} a^{4} - \frac{510027103264661}{35201982478314875} a^{3} + \frac{1}{10} a^{2} - \frac{10710496445887}{90843825750490} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{60}$, which has order $240$ (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:  \( 83086476.1699 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-1271}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-31}, \sqrt{41})\), 4.2.52111.1 x2, 4.0.39401.1 x2, 8.0.2609649624481.2, 8.2.141510355443631.4 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$