↑

Properties

Label 16.8.80291031759...9473.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{15}$
Root discriminant $1315.35$
Ramified primes $37, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55890105463, -171094769281, 12774229580, 304883890488, -122390139278, -37907570302, 15713408910, -2641265466, 663628006, -818830, -1225116, 757494, -126158, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 126158*x^12 + 757494*x^11 - 1225116*x^10 - 818830*x^9 + 663628006*x^8 - 2641265466*x^7 + 15713408910*x^6 - 37907570302*x^5 - 122390139278*x^4 + 304883890488*x^3 + 12774229580*x^2 - 171094769281*x + 55890105463)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 126158*x^12 + 757494*x^11 - 1225116*x^10 - 818830*x^9 + 663628006*x^8 - 2641265466*x^7 + 15713408910*x^6 - 37907570302*x^5 - 122390139278*x^4 + 304883890488*x^3 + 12774229580*x^2 - 171094769281*x + 55890105463, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 126158 x^{12} + 757494 x^{11} - 1225116 x^{10} - 818830 x^{9} + 663628006 x^{8} - 2641265466 x^{7} + 15713408910 x^{6} - 37907570302 x^{5} - 122390139278 x^{4} + 304883890488 x^{3} + 12774229580 x^{2} - 171094769281 x + 55890105463 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(80291031759964036646878404141998486437104406229473=37^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1315.35$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37} a^{8} - \frac{4}{37} a^{7} + \frac{7}{37} a^{6} - \frac{7}{37} a^{5} + \frac{9}{37} a^{4} - \frac{11}{37} a^{3} + \frac{12}{37} a^{2} - \frac{7}{37} a + \frac{12}{37}$, $\frac{1}{37} a^{9} - \frac{9}{37} a^{7} - \frac{16}{37} a^{6} + \frac{18}{37} a^{5} - \frac{12}{37} a^{4} + \frac{5}{37} a^{3} + \frac{4}{37} a^{2} - \frac{16}{37} a + \frac{11}{37}$, $\frac{1}{37} a^{10} - \frac{15}{37} a^{7} + \frac{7}{37} a^{6} - \frac{1}{37} a^{5} + \frac{12}{37} a^{4} + \frac{16}{37} a^{3} + \frac{18}{37} a^{2} - \frac{15}{37} a - \frac{3}{37}$, $\frac{1}{37} a^{11} - \frac{16}{37} a^{7} - \frac{7}{37} a^{6} + \frac{18}{37} a^{5} + \frac{3}{37} a^{4} + \frac{1}{37} a^{3} + \frac{17}{37} a^{2} + \frac{3}{37} a - \frac{5}{37}$, $\frac{1}{851} a^{12} - \frac{6}{851} a^{11} + \frac{4}{851} a^{10} - \frac{11}{851} a^{9} - \frac{9}{851} a^{8} + \frac{174}{851} a^{7} + \frac{128}{851} a^{6} + \frac{347}{851} a^{5} - \frac{403}{851} a^{4} - \frac{94}{851} a^{3} + \frac{87}{851} a^{2} + \frac{81}{851} a - \frac{315}{851}$, $\frac{1}{851} a^{13} - \frac{9}{851} a^{11} - \frac{10}{851} a^{10} - \frac{6}{851} a^{9} + \frac{5}{851} a^{8} + \frac{137}{851} a^{7} - \frac{265}{851} a^{6} - \frac{4}{37} a^{5} - \frac{327}{851} a^{4} - \frac{63}{851} a^{3} + \frac{327}{851} a^{2} + \frac{286}{851} a - \frac{4}{851}$, $\frac{1}{1242065428831643392368301878137841805321} a^{14} - \frac{7}{1242065428831643392368301878137841805321} a^{13} + \frac{243015366608839017864615182475964801}{1242065428831643392368301878137841805321} a^{12} - \frac{6538530043287148462725072174241205}{5569800129289880683265927704653999127} a^{11} - \frac{7508340539603494752061953618743273829}{1242065428831643392368301878137841805321} a^{10} - \frac{16231123967233860925693255147826746467}{1242065428831643392368301878137841805321} a^{9} + \frac{11464214568038262421504184447408365694}{1242065428831643392368301878137841805321} a^{8} + \frac{236906886821279182694748765887076876445}{1242065428831643392368301878137841805321} a^{7} + \frac{524621902774083620113401783779495191890}{1242065428831643392368301878137841805321} a^{6} - \frac{528291637178161145293091268950115475110}{1242065428831643392368301878137841805321} a^{5} - \frac{266011779634268350855682148770839847205}{1242065428831643392368301878137841805321} a^{4} - \frac{278436867559204901894942613638319803159}{1242065428831643392368301878137841805321} a^{3} + \frac{446804471618251967402232543963091666635}{1242065428831643392368301878137841805321} a^{2} + \frac{13665161600724197414109852902884370885}{54002844731810582276882690353819208927} a - \frac{148557401068422973424277069711711425084}{1242065428831643392368301878137841805321}$, $\frac{1}{55459463462761709112637047160732774449387971} a^{15} + \frac{22318}{55459463462761709112637047160732774449387971} a^{14} + \frac{20680413474961855873643562983378810167625}{55459463462761709112637047160732774449387971} a^{13} - \frac{31242612069212604337572814473903973137852}{55459463462761709112637047160732774449387971} a^{12} - \frac{725424671824987978482383298997723944849440}{55459463462761709112637047160732774449387971} a^{11} - \frac{710874982336715343165814394323873796195179}{55459463462761709112637047160732774449387971} a^{10} - \frac{704701601835757247731043511516808898497754}{55459463462761709112637047160732774449387971} a^{9} - \frac{632083726539362411389333428646554466055895}{55459463462761709112637047160732774449387971} a^{8} + \frac{17051077360469937620833742400550495611951349}{55459463462761709112637047160732774449387971} a^{7} - \frac{18911195986289926025053986474409782359345509}{55459463462761709112637047160732774449387971} a^{6} - \frac{21687781160221394380500507737343295942912702}{55459463462761709112637047160732774449387971} a^{5} - \frac{3393565871232364239145423118076711284550668}{55459463462761709112637047160732774449387971} a^{4} - \frac{558874069012346921513457247916760142509305}{1498904417912478624665866139479264174307783} a^{3} - \frac{24977184481346225740131146394994098139992551}{55459463462761709112637047160732774449387971} a^{2} - \frac{769894091724460492858946451247543308907928}{55459463462761709112637047160732774449387971} a + \frac{218676355625543531646210069728589958247263}{55459463462761709112637047160732774449387971}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $11$
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:  \( 834152232017000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.532564273.1, 8.8.28344602131194663732673.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
73Data not computed