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Properties

Label 16.2.77805985826...2851.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 43^{7}$
Root discriminant $73.82$
Ramified primes $17, 43$
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![-239904, 328440, -33830, -92531, -8772, 19125, 9639, 3757, -4845, -1241, 255, 255, 85, -17, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 17*x^14 - 17*x^13 + 85*x^12 + 255*x^11 + 255*x^10 - 1241*x^9 - 4845*x^8 + 3757*x^7 + 9639*x^6 + 19125*x^5 - 8772*x^4 - 92531*x^3 - 33830*x^2 + 328440*x - 239904)
 
gp: K = bnfinit(x^16 - 17*x^14 - 17*x^13 + 85*x^12 + 255*x^11 + 255*x^10 - 1241*x^9 - 4845*x^8 + 3757*x^7 + 9639*x^6 + 19125*x^5 - 8772*x^4 - 92531*x^3 - 33830*x^2 + 328440*x - 239904, 1)
 

Normalized defining polynomial

\( x^{16} - 17 x^{14} - 17 x^{13} + 85 x^{12} + 255 x^{11} + 255 x^{10} - 1241 x^{9} - 4845 x^{8} + 3757 x^{7} + 9639 x^{6} + 19125 x^{5} - 8772 x^{4} - 92531 x^{3} - 33830 x^{2} + 328440 x - 239904 \)

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:  \(-778059858262058848230673812851=-\,17^{15}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.82$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{60} a^{10} + \frac{1}{30} a^{9} + \frac{1}{30} a^{8} + \frac{1}{30} a^{7} + \frac{2}{15} a^{6} + \frac{2}{15} a^{5} - \frac{7}{60} a^{4} - \frac{11}{30} a^{3} - \frac{1}{30} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{120} a^{11} - \frac{1}{60} a^{9} + \frac{1}{40} a^{8} - \frac{1}{20} a^{7} + \frac{1}{10} a^{6} + \frac{9}{40} a^{5} + \frac{1}{10} a^{4} - \frac{29}{60} a^{3} + \frac{11}{40} a^{2} + \frac{13}{60} a - \frac{2}{5}$, $\frac{1}{360} a^{12} - \frac{1}{360} a^{11} - \frac{11}{360} a^{9} - \frac{1}{72} a^{8} + \frac{11}{180} a^{7} - \frac{89}{360} a^{6} + \frac{1}{360} a^{5} - \frac{1}{15} a^{4} - \frac{13}{360} a^{3} - \frac{11}{360} a^{2} - \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{360} a^{13} - \frac{1}{360} a^{11} + \frac{1}{360} a^{10} + \frac{1}{45} a^{9} + \frac{11}{360} a^{8} + \frac{17}{360} a^{7} + \frac{17}{90} a^{6} - \frac{47}{360} a^{5} - \frac{61}{360} a^{4} + \frac{11}{30} a^{3} - \frac{11}{360} a^{2} - \frac{17}{60} a + \frac{2}{5}$, $\frac{1}{181440} a^{14} + \frac{169}{181440} a^{13} - \frac{1}{20160} a^{12} - \frac{691}{181440} a^{11} + \frac{131}{20160} a^{10} + \frac{7517}{181440} a^{9} + \frac{5777}{181440} a^{8} + \frac{1033}{5184} a^{7} - \frac{659}{181440} a^{6} - \frac{35633}{181440} a^{5} + \frac{42809}{181440} a^{4} - \frac{1333}{4032} a^{3} + \frac{3821}{18144} a^{2} + \frac{2389}{7560} a - \frac{31}{90}$, $\frac{1}{3294171342544320} a^{15} - \frac{1263388411}{1647085671272160} a^{14} - \frac{3799276031}{22876189878780} a^{13} + \frac{9755119819}{164708567127216} a^{12} + \frac{22275556565}{9150475951512} a^{11} + \frac{3014153266867}{411771417818040} a^{10} - \frac{68165733109037}{1647085671272160} a^{9} - \frac{18143927825053}{823542835636080} a^{8} + \frac{9436008444979}{82354283563608} a^{7} + \frac{7484325987721}{823542835636080} a^{6} - \frac{18807628721743}{82354283563608} a^{5} - \frac{2624338249031}{45752379757560} a^{4} - \frac{1064418889624519}{3294171342544320} a^{3} - \frac{8022385944445}{109805711418144} a^{2} + \frac{1402836142903}{9150475951512} a + \frac{90518420493}{181557062530}$

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:  \( 4535440485.79 \) (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

\(\Q(\sqrt{17}) \), 4.2.211259.1, 8.2.32624796874211.1

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

Sibling fields

Galois closure: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
43Data not computed