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Properties

Label 16.0.40092926956...969.19
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 47^{8}$
Root discriminant $81.79$
Ramified primes $17, 47$
Class number $5760$ (GRH)
Class group $[4, 12, 120]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82509824, -10122240, 13870848, -8513728, 6950160, -3669248, 1987704, -729716, 281589, -76124, 25776, -6452, 1682, -308, 64, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824)
 
gp: K = bnfinit(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1682 x^{12} - 6452 x^{11} + 25776 x^{10} - 76124 x^{9} + 281589 x^{8} - 729716 x^{7} + 1987704 x^{6} - 3669248 x^{5} + 6950160 x^{4} - 8513728 x^{3} + 13870848 x^{2} - 10122240 x + 82509824 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4009292695690170390860412175969=17^{14}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.79$
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:  $17, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{2176} a^{8} - \frac{1}{544} a^{7} + \frac{29}{1088} a^{6} + \frac{7}{136} a^{5} - \frac{87}{2176} a^{4} + \frac{41}{544} a^{3} - \frac{57}{544} a^{2} - \frac{1}{136} a + \frac{2}{17}$, $\frac{1}{2176} a^{9} + \frac{21}{1088} a^{7} - \frac{1}{34} a^{6} - \frac{47}{2176} a^{5} - \frac{3}{136} a^{4} - \frac{63}{544} a^{3} + \frac{27}{136} a^{2} + \frac{3}{34} a + \frac{8}{17}$, $\frac{1}{4352} a^{10} - \frac{1}{4352} a^{9} - \frac{37}{2176} a^{7} + \frac{29}{4352} a^{6} - \frac{81}{4352} a^{5} + \frac{93}{2176} a^{4} + \frac{47}{1088} a^{3} - \frac{91}{544} a^{2} + \frac{47}{136} a + \frac{5}{17}$, $\frac{1}{4352} a^{11} - \frac{1}{4352} a^{9} - \frac{69}{4352} a^{7} - \frac{7}{272} a^{6} + \frac{233}{4352} a^{5} - \frac{5}{272} a^{4} - \frac{25}{1088} a^{3} - \frac{5}{68} a^{2} - \frac{9}{68} a - \frac{6}{17}$, $\frac{1}{193089536} a^{12} - \frac{3}{96544768} a^{11} + \frac{1509}{193089536} a^{10} - \frac{3745}{96544768} a^{9} + \frac{36375}{193089536} a^{8} - \frac{50313}{96544768} a^{7} - \frac{115953}{11358208} a^{6} + \frac{3117201}{96544768} a^{5} + \frac{2112529}{48272384} a^{4} - \frac{3424169}{24136192} a^{3} - \frac{556097}{3017024} a^{2} + \frac{393285}{1508512} a + \frac{10123}{47141}$, $\frac{1}{193089536} a^{13} + \frac{1473}{193089536} a^{11} + \frac{23}{2839552} a^{10} - \frac{8565}{193089536} a^{9} + \frac{3611}{24136192} a^{8} - \frac{130589}{11358208} a^{7} + \frac{332151}{48272384} a^{6} - \frac{17887}{12068096} a^{5} - \frac{323557}{12068096} a^{4} + \frac{924425}{12068096} a^{3} - \frac{174125}{1508512} a^{2} + \frac{12705}{44368} a + \frac{8051}{47141}$, $\frac{1}{386179072} a^{14} - \frac{1}{386179072} a^{13} - \frac{1}{386179072} a^{12} + \frac{8935}{386179072} a^{11} - \frac{15995}{386179072} a^{10} + \frac{74449}{386179072} a^{9} - \frac{50707}{386179072} a^{8} + \frac{10958573}{386179072} a^{7} - \frac{1217365}{193089536} a^{6} - \frac{4685481}{96544768} a^{5} + \frac{42137}{1027072} a^{4} + \frac{123929}{754256} a^{3} + \frac{36773}{177472} a^{2} + \frac{97335}{377128} a + \frac{585}{2773}$, $\frac{1}{52078951112704} a^{15} + \frac{67421}{52078951112704} a^{14} + \frac{101203}{52078951112704} a^{13} - \frac{17135}{52078951112704} a^{12} - \frac{1185332487}{52078951112704} a^{11} + \frac{4475051079}{52078951112704} a^{10} - \frac{11739904735}{52078951112704} a^{9} - \frac{7485817637}{52078951112704} a^{8} + \frac{569849240177}{26039475556352} a^{7} + \frac{1035857207}{95733366016} a^{6} + \frac{30792575357}{3254934444544} a^{5} - \frac{2435070743}{191466732032} a^{4} + \frac{2060219389}{27584190208} a^{3} + \frac{8842388027}{101716701392} a^{2} - \frac{3696900581}{101716701392} a + \frac{1396154377}{6357293837}$

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

Class group and class number

$C_{4}\times C_{12}\times C_{120}$, which has order $5760$ (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:  $7$
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:  \( 2668614326.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.230911.1 x2, 4.0.10852817.1 x2, 8.0.117783636835489.1, 8.2.42602592046879.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$