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Properties

Label 16.12.1398851876...5937.1
Degree $16$
Signature $[12, 2]$
Discriminant $47^{2}\cdot 97^{15}$
Root discriminant $117.93$
Ramified primes $47, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20855909, 39677155, 53908555, -34561989, -24578465, 9914265, 3584534, -1496025, -135236, 150813, -13138, -10484, 1604, 410, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 68*x^14 + 410*x^13 + 1604*x^12 - 10484*x^11 - 13138*x^10 + 150813*x^9 - 135236*x^8 - 1496025*x^7 + 3584534*x^6 + 9914265*x^5 - 24578465*x^4 - 34561989*x^3 + 53908555*x^2 + 39677155*x - 20855909)
 
gp: K = bnfinit(x^16 - 6*x^15 - 68*x^14 + 410*x^13 + 1604*x^12 - 10484*x^11 - 13138*x^10 + 150813*x^9 - 135236*x^8 - 1496025*x^7 + 3584534*x^6 + 9914265*x^5 - 24578465*x^4 - 34561989*x^3 + 53908555*x^2 + 39677155*x - 20855909, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 68 x^{14} + 410 x^{13} + 1604 x^{12} - 10484 x^{11} - 13138 x^{10} + 150813 x^{9} - 135236 x^{8} - 1496025 x^{7} + 3584534 x^{6} + 9914265 x^{5} - 24578465 x^{4} - 34561989 x^{3} + 53908555 x^{2} + 39677155 x - 20855909 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1398851876803167753769596583695937=47^{2}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.93$
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:  $47, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{803382453566295156430048899081275859355916539517399191} a^{15} + \frac{144066488622081468528135051593544544685178253923681141}{803382453566295156430048899081275859355916539517399191} a^{14} - \frac{326168214189745186205093923595403279746718265373315136}{803382453566295156430048899081275859355916539517399191} a^{13} + \frac{251379350346465607013620893327706121030252824882011033}{803382453566295156430048899081275859355916539517399191} a^{12} + \frac{57169351595578273376407809661384101436723438262058595}{803382453566295156430048899081275859355916539517399191} a^{11} + \frac{297976660643022899716858605339684576818989542922425963}{803382453566295156430048899081275859355916539517399191} a^{10} + \frac{1000946147344379586192278537953244486626071364695355}{7109579235099957136549105301604211144742624243516807} a^{9} + \frac{370078044661069268628269265652647772639097481392399264}{803382453566295156430048899081275859355916539517399191} a^{8} + \frac{12651787588341106162896158867796160152825692206896765}{803382453566295156430048899081275859355916539517399191} a^{7} - \frac{297106339583669850420554415818824538830730394995412319}{803382453566295156430048899081275859355916539517399191} a^{6} - \frac{271060999590826202948066922654652983084575500112979381}{803382453566295156430048899081275859355916539517399191} a^{5} - \frac{328214133285879907496450627582006358162833305763823361}{803382453566295156430048899081275859355916539517399191} a^{4} - \frac{272678393458966840411550087968824002738197570218023046}{803382453566295156430048899081275859355916539517399191} a^{3} - \frac{371796245760213053665039115825728310585339502895268323}{803382453566295156430048899081275859355916539517399191} a^{2} - \frac{148684314445547512922898710889954296362820058133877356}{803382453566295156430048899081275859355916539517399191} a - \frac{65261815107776510469144683603125536659202676942928572}{803382453566295156430048899081275859355916539517399191}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 38504780186.6 \) (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$47$47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed