↑

Properties

Label 16.0.22005987491...0176.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 41^{14}\cdot 97^{4}$
Root discriminant $161.77$
Ramified primes $2, 41, 97$
Class number $433108$ (GRH)
Class group $[2, 216554]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T209)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4014327025, -1515380265, 1719758063, -979188614, 500688278, -193059815, 55333563, -17495412, 4134399, -1057485, 255841, -30868, 9346, -652, 174, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 174*x^14 - 652*x^13 + 9346*x^12 - 30868*x^11 + 255841*x^10 - 1057485*x^9 + 4134399*x^8 - 17495412*x^7 + 55333563*x^6 - 193059815*x^5 + 500688278*x^4 - 979188614*x^3 + 1719758063*x^2 - 1515380265*x + 4014327025)
 
gp: K = bnfinit(x^16 - 6*x^15 + 174*x^14 - 652*x^13 + 9346*x^12 - 30868*x^11 + 255841*x^10 - 1057485*x^9 + 4134399*x^8 - 17495412*x^7 + 55333563*x^6 - 193059815*x^5 + 500688278*x^4 - 979188614*x^3 + 1719758063*x^2 - 1515380265*x + 4014327025, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 174 x^{14} - 652 x^{13} + 9346 x^{12} - 30868 x^{11} + 255841 x^{10} - 1057485 x^{9} + 4134399 x^{8} - 17495412 x^{7} + 55333563 x^{6} - 193059815 x^{5} + 500688278 x^{4} - 979188614 x^{3} + 1719758063 x^{2} - 1515380265 x + 4014327025 \)

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:  \(220059874915222898374641881988530176=2^{16}\cdot 41^{14}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.77$
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:  $2, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{15} + \frac{1277422479143761105312994616382257053788397894286886101973597309782565709}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{14} - \frac{1099020498313945552989542449155008956611713235521972101350545297353479381}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{13} - \frac{21764346252557180889573598842129230422087443236005646053763810795107527}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{12} - \frac{574718427441995387316169763142252229489805737208446949365711689402178424}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{11} + \frac{1293123819701916144768667906012008856213029290214393614977762267003178627}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{10} - \frac{391452761884002931295705125667476984892478023904530858193248520600836279}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{9} + \frac{224241583775121010564457793772220616661649405634517174936745923442496933}{686559515616169222655368716455935404049456075936591161778283714847127181} a^{8} - \frac{1204403754124115703406155452902513018150947103501352316495402655717483246}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{7} + \frac{1454430724096979086927132536439309108057491241243016981164136643762908418}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{6} + \frac{1629549980629984766315284523670700193914753385910587940526748298148867103}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{5} - \frac{184224270759715679657574161505704283619935161331325186167061946179325476}{686559515616169222655368716455935404049456075936591161778283714847127181} a^{4} - \frac{979259441187634282044887443596425179602147646747304810101622818601260712}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{3} + \frac{1214736996949551813398197157636648797585050469779283549811529378112635131}{3432797578080846113276843582279677020247280379682955808891418574235635905} a^{2} - \frac{1318977796624118799364773802435818515274225587968613166786996325166378642}{3432797578080846113276843582279677020247280379682955808891418574235635905} a + \frac{327176320334747828216542131974185506810047639212034220281858883989691735}{686559515616169222655368716455935404049456075936591161778283714847127181}$

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

Class group and class number

$C_{2}\times C_{216554}$, which has order $433108$ (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:  \( 239630.30249 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T209):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.13448.1, 4.4.68921.1, 4.4.551368.1, 8.8.304006671424.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
41Data not computed
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$