↑

Properties

Label 16.0.45259126231...3889.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 101^{14}$
Root discriminant $535.16$
Ramified primes $13, 101$
Class number $7299856$ (GRH)
Class group $[2, 2, 1824964]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30833678218877, -17032173590210, 2727122691361, -653503073861, 142759809978, 9793290973, 9472812670, 422045666, 178861490, -4367626, 677985, -251782, 123, -2105, 120, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 120*x^14 - 2105*x^13 + 123*x^12 - 251782*x^11 + 677985*x^10 - 4367626*x^9 + 178861490*x^8 + 422045666*x^7 + 9472812670*x^6 + 9793290973*x^5 + 142759809978*x^4 - 653503073861*x^3 + 2727122691361*x^2 - 17032173590210*x + 30833678218877)
 
gp: K = bnfinit(x^16 - x^15 + 120*x^14 - 2105*x^13 + 123*x^12 - 251782*x^11 + 677985*x^10 - 4367626*x^9 + 178861490*x^8 + 422045666*x^7 + 9472812670*x^6 + 9793290973*x^5 + 142759809978*x^4 - 653503073861*x^3 + 2727122691361*x^2 - 17032173590210*x + 30833678218877, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 120 x^{14} - 2105 x^{13} + 123 x^{12} - 251782 x^{11} + 677985 x^{10} - 4367626 x^{9} + 178861490 x^{8} + 422045666 x^{7} + 9472812670 x^{6} + 9793290973 x^{5} + 142759809978 x^{4} - 653503073861 x^{3} + 2727122691361 x^{2} - 17032173590210 x + 30833678218877 \)

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:  \(45259126231720831581016789061101452813693889=13^{14}\cdot 101^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $535.16$
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:  $13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} - \frac{5}{17} a^{10} + \frac{8}{17} a^{7} + \frac{1}{17} a^{6} + \frac{6}{17} a^{5} + \frac{6}{17} a^{4} - \frac{5}{17} a^{3} - \frac{4}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{17} a^{12} - \frac{8}{17} a^{10} + \frac{8}{17} a^{8} + \frac{7}{17} a^{7} - \frac{6}{17} a^{6} + \frac{2}{17} a^{5} + \frac{8}{17} a^{4} + \frac{5}{17} a^{3} + \frac{6}{17} a^{2} - \frac{6}{17} a$, $\frac{1}{17} a^{13} - \frac{6}{17} a^{10} + \frac{8}{17} a^{9} + \frac{7}{17} a^{8} + \frac{7}{17} a^{7} - \frac{7}{17} a^{6} + \frac{5}{17} a^{5} + \frac{2}{17} a^{4} - \frac{4}{17} a^{2} + \frac{4}{17} a$, $\frac{1}{17} a^{14} - \frac{5}{17} a^{10} + \frac{7}{17} a^{9} + \frac{7}{17} a^{8} + \frac{7}{17} a^{7} - \frac{6}{17} a^{6} + \frac{4}{17} a^{5} + \frac{2}{17} a^{4} - \frac{3}{17} a^{2} + \frac{3}{17} a$, $\frac{1}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{15} + \frac{9482322855203639389052916210918048292144769427872894963526586717570860420021072661878599}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{14} + \frac{62975144134424984914656268603532685677029197158263569185412890715691238236896198837394245}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{13} + \frac{31111010733167206970654060476419484860624023196248125699147717244515619316058578418166756}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{12} - \frac{46550529700765198753359033117649075823655630606497642126812480328199373843043001685032455}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{11} + \frac{545191826965095113164342428810790434740014219830630773921950431680552166301912785837476189}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{10} - \frac{1125923825502436919682325058272647325374290383873188762814741212902698632551220828113837359}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{9} - \frac{1321066183228209915369683822495074483482255478826913943990486774762117718231334957181063240}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{8} + \frac{494037429189305137853035232511912710900289332743269991553472544428603610505233472752568134}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{7} + \frac{886070751092084150961634418737430080307853512854972072266178041291188471324987225105615480}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{6} + \frac{1170998093469169041641742141854805271192286678361024899066654337279621777120907893034425144}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{5} - \frac{266899883780834352062837068058547533868472784638069595866424140579980445199606357499963}{1407482578848652496229376574457377250357433073101243218629958452086275101492523773829031} a^{4} + \frac{770645927150451326380574339615029663080498758865169566095353587074802580137074331937605827}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{3} + \frac{1138269282664617285208139143253295830646390988265972360821543923589158538058187923571571758}{2658734591445104565377292349149985625925191075088248439991991515990973666719377408763039559} a^{2} - \frac{21938496437770998556301522656199504348395609232121139504476534977341530254386358285956633}{156396152437947327375134844067646213289717122064014614117175971528880803924669259339002327} a - \frac{255235783855733360619794338262180233590403428097930870625591499227222730172220564011495}{9199773672820431022066755533390953722924536592000859653951527736992988466157015255235431}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1824964}$, which has order $7299856$ (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:  \( 303440268.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{1313}) \), \(\Q(\sqrt{13}, \sqrt{101})\), 8.8.5123755016602262209.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$