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Properties

Label 16.4.12746025680...9401.2
Degree $16$
Signature $[4, 6]$
Discriminant $31^{10}\cdot 41^{15}$
Root discriminant $278.03$
Ramified primes $31, 41$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-209580401, 1790386599, -2009102852, 576210513, -8721981, 19815774, -50674969, 15418771, -4451483, 222254, -242491, -7359, -611, -510, 104, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 104*x^14 - 510*x^13 - 611*x^12 - 7359*x^11 - 242491*x^10 + 222254*x^9 - 4451483*x^8 + 15418771*x^7 - 50674969*x^6 + 19815774*x^5 - 8721981*x^4 + 576210513*x^3 - 2009102852*x^2 + 1790386599*x - 209580401)
 
gp: K = bnfinit(x^16 - 6*x^15 + 104*x^14 - 510*x^13 - 611*x^12 - 7359*x^11 - 242491*x^10 + 222254*x^9 - 4451483*x^8 + 15418771*x^7 - 50674969*x^6 + 19815774*x^5 - 8721981*x^4 + 576210513*x^3 - 2009102852*x^2 + 1790386599*x - 209580401, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 104 x^{14} - 510 x^{13} - 611 x^{12} - 7359 x^{11} - 242491 x^{10} + 222254 x^{9} - 4451483 x^{8} + 15418771 x^{7} - 50674969 x^{6} + 19815774 x^{5} - 8721981 x^{4} + 576210513 x^{3} - 2009102852 x^{2} + 1790386599 x - 209580401 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1274602568003244304847660149105336749401=31^{10}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.03$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{113} a^{14} - \frac{38}{113} a^{13} - \frac{54}{113} a^{12} - \frac{19}{113} a^{11} - \frac{48}{113} a^{10} + \frac{56}{113} a^{9} - \frac{17}{113} a^{8} - \frac{29}{113} a^{7} + \frac{30}{113} a^{6} + \frac{48}{113} a^{5} - \frac{48}{113} a^{4} - \frac{25}{113} a^{3} + \frac{6}{113} a^{2} + \frac{41}{113} a + \frac{20}{113}$, $\frac{1}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{15} + \frac{151632115242083645257500729659053759151610032828592156316223332101902}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{14} + \frac{28250486961778640551919676723627836185196602708984710067627836061557866}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{13} - \frac{5428975095683617012879858709895990070940367945597158798361976555888570}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{12} - \frac{1040427366759551103443126515559727241028935410602756189943156454420919}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{11} + \frac{22896720114357012338645733476580184847376382936815523379140784656916146}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{10} - \frac{28963875041057053526338899694151851561754735419209164964163101165559879}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{9} - \frac{20760670518980220224887443066097296572433394922857490155841631892503749}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{8} - \frac{18005228920398883148582122861754597641372769482465592871789545603312117}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{7} + \frac{13223684560497657964910688278132425391151895667959594592247288698264722}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{6} - \frac{1636785755308698834262738125789419272530259346215806125375043292712787}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{5} + \frac{10295831454816783187617626050340964045315614850921757679901242376843060}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{4} + \frac{9262198256047684426444553792135529324489514182450996168133180832487720}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{3} - \frac{17658420250180380547657298843391477632805018775409252362289268371614759}{65543073171630189692797011580191697451255262009726427350226270579975879} a^{2} + \frac{4377492201946378995149669363834429322506594769611031392532349265023156}{65543073171630189692797011580191697451255262009726427350226270579975879} a - \frac{6149596700804350517290064086888441820228908690479060247930195575600535}{65543073171630189692797011580191697451255262009726427350226270579975879}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $9$
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:  \( 1563337889260 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed