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Properties

Label 16.2.77674857879...9211.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 83^{7}$
Root discriminant $98.43$
Ramified primes $17, 83$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4820096, 2882976, 1183464, -2883720, -1210880, 197569, 409868, 87641, -123455, 59894, -13133, 1659, 4, -29, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 29*x^13 + 4*x^12 + 1659*x^11 - 13133*x^10 + 59894*x^9 - 123455*x^8 + 87641*x^7 + 409868*x^6 + 197569*x^5 - 1210880*x^4 - 2883720*x^3 + 1183464*x^2 + 2882976*x + 4820096)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 29*x^13 + 4*x^12 + 1659*x^11 - 13133*x^10 + 59894*x^9 - 123455*x^8 + 87641*x^7 + 409868*x^6 + 197569*x^5 - 1210880*x^4 - 2883720*x^3 + 1183464*x^2 + 2882976*x + 4820096, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 29 x^{13} + 4 x^{12} + 1659 x^{11} - 13133 x^{10} + 59894 x^{9} - 123455 x^{8} + 87641 x^{7} + 409868 x^{6} + 197569 x^{5} - 1210880 x^{4} - 2883720 x^{3} + 1183464 x^{2} + 2882976 x + 4820096 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-77674857879654074046142123779211=-\,17^{15}\cdot 83^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.43$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{40} a^{11} + \frac{1}{40} a^{10} - \frac{3}{40} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{3}{40} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{40} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{40} a^{13} + \frac{1}{40} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{8} a^{7} - \frac{1}{5} a^{6} - \frac{1}{40} a^{4} - \frac{1}{20} a^{3} - \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{9440} a^{14} + \frac{63}{9440} a^{13} - \frac{31}{4720} a^{12} - \frac{71}{9440} a^{11} - \frac{107}{9440} a^{10} - \frac{3}{1180} a^{9} + \frac{459}{9440} a^{8} - \frac{1819}{9440} a^{7} + \frac{23}{944} a^{6} + \frac{939}{9440} a^{5} + \frac{323}{1888} a^{4} - \frac{261}{1180} a^{3} + \frac{351}{1180} a^{2} - \frac{107}{1180} a + \frac{4}{59}$, $\frac{1}{8690393507671818628633410045116611611222400} a^{15} + \frac{369297029716718406411943769632415839563}{8690393507671818628633410045116611611222400} a^{14} - \frac{26076496916295902503976429825120247819067}{4345196753835909314316705022558305805611200} a^{13} - \frac{20707943979528418404339301089106232030131}{1738078701534363725726682009023322322244480} a^{12} + \frac{184235992015034343305757303096428403762029}{8690393507671818628633410045116611611222400} a^{11} - \frac{19154653806120210341965030351756118253931}{434519675383590931431670502255830580561120} a^{10} - \frac{625608942485006420506095071109857301666173}{8690393507671818628633410045116611611222400} a^{9} - \frac{737171178462104583553507242628470265706263}{8690393507671818628633410045116611611222400} a^{8} - \frac{850868733536312111104526263569546702398921}{4345196753835909314316705022558305805611200} a^{7} - \frac{644165462909043696687639339242486386405597}{8690393507671818628633410045116611611222400} a^{6} - \frac{82113294612222332782078243353991411213441}{347615740306872745145336401804664464448896} a^{5} - \frac{1712146387005632457260697192889746074427}{20304657728205183711760303843730400960800} a^{4} - \frac{122608102814629124260035143478034351979127}{271574797114744332144794063909894112850700} a^{3} + \frac{177437742939403113619761519066775960039363}{1086299188458977328579176255639576451402800} a^{2} - \frac{219028429968702388747668840090124620731}{5431495942294886642895881278197882257014} a - \frac{30955484265725868974471339211128543059078}{67893699278686083036198515977473528212675}$

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:  $8$
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:  \( 29155805986.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.407779.1, 8.2.234626318818651.2

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
17Data not computed
83Data not computed