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Properties

Label 16.2.17348320607...6139.8
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 67^{7}$
Root discriminant $89.63$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[8]$ (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![4608, -91968, 637232, -2111188, 3626014, -3134921, 1123050, -56499, 4489, 988, 5529, -3511, 934, -189, 47, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 47*x^14 - 189*x^13 + 934*x^12 - 3511*x^11 + 5529*x^10 + 988*x^9 + 4489*x^8 - 56499*x^7 + 1123050*x^6 - 3134921*x^5 + 3626014*x^4 - 2111188*x^3 + 637232*x^2 - 91968*x + 4608)
 
gp: K = bnfinit(x^16 - 8*x^15 + 47*x^14 - 189*x^13 + 934*x^12 - 3511*x^11 + 5529*x^10 + 988*x^9 + 4489*x^8 - 56499*x^7 + 1123050*x^6 - 3134921*x^5 + 3626014*x^4 - 2111188*x^3 + 637232*x^2 - 91968*x + 4608, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 47 x^{14} - 189 x^{13} + 934 x^{12} - 3511 x^{11} + 5529 x^{10} + 988 x^{9} + 4489 x^{8} - 56499 x^{7} + 1123050 x^{6} - 3134921 x^{5} + 3626014 x^{4} - 2111188 x^{3} + 637232 x^{2} - 91968 x + 4608 \)

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:  \(-17348320607629615993685048266139=-\,17^{15}\cdot 67^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.63$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{720} a^{10} - \frac{1}{144} a^{9} - \frac{1}{360} a^{8} + \frac{19}{360} a^{7} + \frac{17}{360} a^{6} + \frac{13}{90} a^{5} + \frac{157}{720} a^{4} + \frac{167}{720} a^{3} - \frac{59}{120} a^{2} - \frac{7}{36} a + \frac{1}{15}$, $\frac{1}{720} a^{11} - \frac{3}{80} a^{9} + \frac{7}{180} a^{8} + \frac{11}{180} a^{7} - \frac{43}{360} a^{6} + \frac{137}{720} a^{5} + \frac{13}{180} a^{4} - \frac{239}{720} a^{3} - \frac{29}{72} a^{2} - \frac{73}{180} a + \frac{1}{3}$, $\frac{1}{84240} a^{12} - \frac{1}{14040} a^{11} + \frac{1}{1755} a^{10} - \frac{37}{16848} a^{9} + \frac{1723}{42120} a^{8} + \frac{8}{81} a^{7} + \frac{3323}{84240} a^{6} + \frac{1285}{8424} a^{5} - \frac{629}{21060} a^{4} + \frac{18229}{84240} a^{3} - \frac{4301}{42120} a^{2} - \frac{2909}{7020} a - \frac{29}{117}$, $\frac{1}{84240} a^{13} + \frac{1}{7020} a^{11} - \frac{7}{42120} a^{10} + \frac{2921}{84240} a^{9} + \frac{115}{8424} a^{8} - \frac{7363}{84240} a^{7} + \frac{73}{8424} a^{6} + \frac{391}{5265} a^{5} - \frac{23}{1620} a^{4} + \frac{25073}{84240} a^{3} + \frac{1843}{14040} a^{2} + \frac{229}{780} a + \frac{29}{65}$, $\frac{1}{1518112290240} a^{14} - \frac{1}{216873184320} a^{13} - \frac{116297}{189764036280} a^{12} + \frac{5582347}{1518112290240} a^{11} + \frac{65551459}{216873184320} a^{10} - \frac{1172736373}{759056145120} a^{9} + \frac{29526756299}{1518112290240} a^{8} - \frac{103972782617}{1518112290240} a^{7} - \frac{2209022813}{94882018140} a^{6} - \frac{299030702587}{1518112290240} a^{5} + \frac{55928342881}{506037430080} a^{4} + \frac{49875538483}{253018715040} a^{3} + \frac{26233811147}{75905614512} a^{2} - \frac{1210440475}{3162733938} a + \frac{86932141}{2635611615}$, $\frac{1}{153329341314240} a^{15} + \frac{43}{153329341314240} a^{14} + \frac{49006211}{8518296739680} a^{13} + \frac{589808207}{153329341314240} a^{12} - \frac{207064321}{3407318695872} a^{11} + \frac{901782163}{6388722554760} a^{10} - \frac{13547306113}{2358912943296} a^{9} + \frac{1845675825197}{153329341314240} a^{8} - \frac{716067691}{50404122720} a^{7} + \frac{841632929621}{30665868262848} a^{6} - \frac{4649365109837}{21904191616320} a^{5} + \frac{35982842191}{182534930136} a^{4} + \frac{181199258839}{9583083832140} a^{3} - \frac{7986874101263}{19166167664280} a^{2} - \frac{141695726027}{798590319345} a - \frac{35223609727}{266196773115}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 12490221748.4 \) (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

\(\Q(\sqrt{17}) \), 4.2.329171.1, 8.2.123414690307499.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
67Data not computed