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Properties

Label 16.0.88848937405...8553.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 73^{13}$
Root discriminant $74.44$
Ramified primes $3, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52726789, 171975278, 244084510, 193906446, 87471399, 16998374, -1709406, -670143, 363862, 91982, -24509, -4394, 1710, 201, -46, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 46*x^14 + 201*x^13 + 1710*x^12 - 4394*x^11 - 24509*x^10 + 91982*x^9 + 363862*x^8 - 670143*x^7 - 1709406*x^6 + 16998374*x^5 + 87471399*x^4 + 193906446*x^3 + 244084510*x^2 + 171975278*x + 52726789)
 
gp: K = bnfinit(x^16 - x^15 - 46*x^14 + 201*x^13 + 1710*x^12 - 4394*x^11 - 24509*x^10 + 91982*x^9 + 363862*x^8 - 670143*x^7 - 1709406*x^6 + 16998374*x^5 + 87471399*x^4 + 193906446*x^3 + 244084510*x^2 + 171975278*x + 52726789, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 46 x^{14} + 201 x^{13} + 1710 x^{12} - 4394 x^{11} - 24509 x^{10} + 91982 x^{9} + 363862 x^{8} - 670143 x^{7} - 1709406 x^{6} + 16998374 x^{5} + 87471399 x^{4} + 193906446 x^{3} + 244084510 x^{2} + 171975278 x + 52726789 \)

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:  \(888489374058862559333388518553=3^{12}\cdot 73^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.44$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{12} - \frac{3}{14} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{14} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{5}{14} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{686} a^{13} - \frac{17}{686} a^{12} + \frac{5}{343} a^{11} - \frac{3}{686} a^{10} - \frac{195}{686} a^{9} + \frac{3}{49} a^{8} - \frac{39}{686} a^{7} + \frac{291}{686} a^{6} + \frac{86}{343} a^{5} - \frac{83}{686} a^{4} + \frac{11}{98} a^{3} + \frac{157}{343} a^{2} + \frac{249}{686} a - \frac{73}{686}$, $\frac{1}{4802} a^{14} + \frac{113}{4802} a^{12} - \frac{29}{4802} a^{11} - \frac{1152}{2401} a^{10} + \frac{353}{4802} a^{9} - \frac{501}{4802} a^{8} + \frac{59}{2401} a^{7} + \frac{317}{4802} a^{6} + \frac{1665}{4802} a^{5} - \frac{1010}{2401} a^{4} + \frac{643}{4802} a^{3} - \frac{979}{4802} a^{2} + \frac{365}{2401} a + \frac{1699}{4802}$, $\frac{1}{2932444673533221479979138668886358140811702373522} a^{15} + \frac{35470276530376423407834386703972246608882515}{1466222336766610739989569334443179070405851186761} a^{14} + \frac{389633470388010874857342723173288828120895825}{1466222336766610739989569334443179070405851186761} a^{13} - \frac{21495517511472910748552072713519975708677752788}{1466222336766610739989569334443179070405851186761} a^{12} - \frac{59270670416420260273874244725794322481806638202}{1466222336766610739989569334443179070405851186761} a^{11} - \frac{98522953481672034980544237873985118520178452800}{209460333823801534284224190634739867200835883823} a^{10} + \frac{265902981700244381902565350935877883361479296599}{1466222336766610739989569334443179070405851186761} a^{9} - \frac{58684723902655456521102782533410880835735270301}{1466222336766610739989569334443179070405851186761} a^{8} - \frac{279397853172911024593291275066447530847625154690}{1466222336766610739989569334443179070405851186761} a^{7} + \frac{724946613249323365498680016271005192891230551043}{1466222336766610739989569334443179070405851186761} a^{6} - \frac{65415044404796777702527175996860138723983582766}{209460333823801534284224190634739867200835883823} a^{5} + \frac{179641103114731852222707406905355242177945031581}{1466222336766610739989569334443179070405851186761} a^{4} - \frac{421853304195063039344463646870855406349603239850}{1466222336766610739989569334443179070405851186761} a^{3} + \frac{507130639952605657943458607191416762627326834091}{1466222336766610739989569334443179070405851186761} a^{2} + \frac{134770554847192210097100513498657258147519859037}{1466222336766610739989569334443179070405851186761} a + \frac{1441207422217304340561029389744743994503747391707}{2932444673533221479979138668886358140811702373522}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{65908090403000873002117768476}{59564524585773750681730461592764091} a^{15} - \frac{114769637318812829413745304836}{59564524585773750681730461592764091} a^{14} - \frac{3012338388142945471333619242301}{59564524585773750681730461592764091} a^{13} + \frac{15790080921332858261261616030521}{59564524585773750681730461592764091} a^{12} + \frac{103083970668727903956542359189944}{59564524585773750681730461592764091} a^{11} - \frac{387953221952956007224090205884120}{59564524585773750681730461592764091} a^{10} - \frac{1365676293551380896973885132534974}{59564524585773750681730461592764091} a^{9} + \frac{7575997703024941563252716694791346}{59564524585773750681730461592764091} a^{8} + \frac{18250246830818473863411956190684547}{59564524585773750681730461592764091} a^{7} - \frac{64982773509076867111670822221118402}{59564524585773750681730461592764091} a^{6} - \frac{61157261301212834421171147594016501}{59564524585773750681730461592764091} a^{5} + \frac{1222241474068323507215709015400985579}{59564524585773750681730461592764091} a^{4} + \frac{4752226755114898827316182574942700579}{59564524585773750681730461592764091} a^{3} + \frac{8340627074692923594364674386504859446}{59564524585773750681730461592764091} a^{2} + \frac{7907130331792393887310858640367824173}{59564524585773750681730461592764091} a + \frac{3303445290857451040591528007299965025}{59564524585773750681730461592764091} \) (order $6$)
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:  \( 381368738.095 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.657.1, 8.0.167918799033.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $16$ $16$ $16$

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
3Data not computed
$73$73.8.7.7$x^{8} + 228125$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.6.3$x^{8} - 73 x^{4} + 58619$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$