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Properties

Label 16.0.23302078379...3161.1
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 29^{14}$
Root discriminant $91.30$
Ramified primes $23, 29$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9032369, 6921894, 14900229, 5086513, 5779990, -41789, 594500, -372244, 49958, -54366, 11703, -3600, 1215, -194, 52, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 52*x^14 - 194*x^13 + 1215*x^12 - 3600*x^11 + 11703*x^10 - 54366*x^9 + 49958*x^8 - 372244*x^7 + 594500*x^6 - 41789*x^5 + 5779990*x^4 + 5086513*x^3 + 14900229*x^2 + 6921894*x + 9032369)
 
gp: K = bnfinit(x^16 - 6*x^15 + 52*x^14 - 194*x^13 + 1215*x^12 - 3600*x^11 + 11703*x^10 - 54366*x^9 + 49958*x^8 - 372244*x^7 + 594500*x^6 - 41789*x^5 + 5779990*x^4 + 5086513*x^3 + 14900229*x^2 + 6921894*x + 9032369, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 52 x^{14} - 194 x^{13} + 1215 x^{12} - 3600 x^{11} + 11703 x^{10} - 54366 x^{9} + 49958 x^{8} - 372244 x^{7} + 594500 x^{6} - 41789 x^{5} + 5779990 x^{4} + 5086513 x^{3} + 14900229 x^{2} + 6921894 x + 9032369 \)

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:  \(23302078379314905029568288023161=23^{8}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.30$
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:  $23, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{805} a^{12} + \frac{7}{115} a^{11} - \frac{10}{161} a^{10} + \frac{272}{805} a^{9} - \frac{3}{115} a^{8} - \frac{267}{805} a^{7} + \frac{363}{805} a^{6} - \frac{12}{161} a^{5} - \frac{129}{805} a^{4} + \frac{27}{805} a^{3} - \frac{44}{115} a^{2} - \frac{94}{805} a + \frac{187}{805}$, $\frac{1}{805} a^{13} - \frac{36}{805} a^{11} - \frac{3}{161} a^{10} + \frac{5}{23} a^{9} + \frac{279}{805} a^{8} + \frac{83}{805} a^{7} - \frac{298}{805} a^{6} + \frac{47}{161} a^{5} + \frac{17}{35} a^{4} - \frac{3}{115} a^{3} + \frac{347}{805} a^{2} + \frac{57}{161} a + \frac{2}{115}$, $\frac{1}{467705} a^{14} + \frac{82}{467705} a^{13} - \frac{48}{467705} a^{12} + \frac{5139}{467705} a^{11} - \frac{1836}{66815} a^{10} - \frac{78151}{467705} a^{9} + \frac{187433}{467705} a^{8} - \frac{70627}{467705} a^{7} + \frac{186378}{467705} a^{6} - \frac{60602}{467705} a^{5} + \frac{8843}{93541} a^{4} + \frac{133863}{467705} a^{3} - \frac{232571}{467705} a^{2} + \frac{107427}{467705} a + \frac{62338}{467705}$, $\frac{1}{114325250336505704426028322913032154615} a^{15} - \frac{78802051586700202525755532416}{101442103226713136136671093977845745} a^{14} - \frac{15579855831851505314242378636505527}{114325250336505704426028322913032154615} a^{13} + \frac{55553308264086454411682204687136971}{114325250336505704426028322913032154615} a^{12} - \frac{8677962072326929297117200041860740026}{114325250336505704426028322913032154615} a^{11} - \frac{8788680292608497353766800225170168106}{114325250336505704426028322913032154615} a^{10} + \frac{1684310736346531047939745881161937923}{114325250336505704426028322913032154615} a^{9} - \frac{42158285303916191962548276867793543283}{114325250336505704426028322913032154615} a^{8} - \frac{28948811883109023125472582575477606323}{114325250336505704426028322913032154615} a^{7} + \frac{50391952985640805426184728514747741129}{114325250336505704426028322913032154615} a^{6} + \frac{21154424526221728826181332055271208531}{114325250336505704426028322913032154615} a^{5} + \frac{8607714137450638564839370716354009921}{114325250336505704426028322913032154615} a^{4} - \frac{25754925879870112966605741243979578309}{114325250336505704426028322913032154615} a^{3} - \frac{3820294442187554036887280254789634976}{22865050067301140885205664582606430923} a^{2} - \frac{154709945209740551919250447839447299}{22865050067301140885205664582606430923} a - \frac{2099098523246013440008871744512851415}{22865050067301140885205664582606430923}$

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:  \( 222324222.161 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-667}) \), \(\Q(\sqrt{-23}) \), 4.0.24389.1, 4.4.12901781.1, \(\Q(\sqrt{-23}, \sqrt{29})\), 8.0.9125184567461.1 x2, 8.4.4827222636186869.1 x2, 8.0.166455952971961.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$