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Properties

Label 16.0.50321575323...2193.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{11}\cdot 59^{8}$
Root discriminant $53.87$
Ramified primes $17, 59$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![680627, 1086394, 1251998, 926259, 452160, 137244, -2902, -21167, -9712, -2412, 911, 371, 43, 13, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 13*x^14 + 13*x^13 + 43*x^12 + 371*x^11 + 911*x^10 - 2412*x^9 - 9712*x^8 - 21167*x^7 - 2902*x^6 + 137244*x^5 + 452160*x^4 + 926259*x^3 + 1251998*x^2 + 1086394*x + 680627)
 
gp: K = bnfinit(x^16 - 2*x^15 - 13*x^14 + 13*x^13 + 43*x^12 + 371*x^11 + 911*x^10 - 2412*x^9 - 9712*x^8 - 21167*x^7 - 2902*x^6 + 137244*x^5 + 452160*x^4 + 926259*x^3 + 1251998*x^2 + 1086394*x + 680627, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 13 x^{14} + 13 x^{13} + 43 x^{12} + 371 x^{11} + 911 x^{10} - 2412 x^{9} - 9712 x^{8} - 21167 x^{7} - 2902 x^{6} + 137244 x^{5} + 452160 x^{4} + 926259 x^{3} + 1251998 x^{2} + 1086394 x + 680627 \)

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:  \(5032157532379666474146082193=17^{11}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.87$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4651130553864586533799951815541383633750299} a^{15} + \frac{2252279574020360633763915615300925165754605}{4651130553864586533799951815541383633750299} a^{14} - \frac{300693035658017294455829716221264752842352}{4651130553864586533799951815541383633750299} a^{13} - \frac{531852541810872634528348353685951739842527}{4651130553864586533799951815541383633750299} a^{12} - \frac{1675656728404772271497781401007557118044963}{4651130553864586533799951815541383633750299} a^{11} + \frac{1196495328381757936183391198860468128859021}{4651130553864586533799951815541383633750299} a^{10} + \frac{1880061775023338361314912825102948494918473}{4651130553864586533799951815541383633750299} a^{9} - \frac{473899317930473385779699896191648846814106}{4651130553864586533799951815541383633750299} a^{8} - \frac{230730000013366032101949663398393179846225}{4651130553864586533799951815541383633750299} a^{7} + \frac{585644743568301536381594529911075126260568}{4651130553864586533799951815541383633750299} a^{6} + \frac{2255075826338449452808199478678162552117978}{4651130553864586533799951815541383633750299} a^{5} - \frac{274337669409960108504420581970455997417143}{4651130553864586533799951815541383633750299} a^{4} - \frac{1207853639093382974406515320165409605206677}{4651130553864586533799951815541383633750299} a^{3} - \frac{1690224798338862809792016128630361474909093}{4651130553864586533799951815541383633750299} a^{2} - \frac{83127333870458567440233292590996200113260}{4651130553864586533799951815541383633750299} a + \frac{1024863617625704104240025550237317952428134}{4651130553864586533799951815541383633750299}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 1821925.34206 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.59177.1, 8.0.59532594593.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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$17.8.7.5$x^{8} + 459$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
59Data not computed