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Properties

Label 16.0.93675802669...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}$
Root discriminant $48.50$
Ramified primes $5, 13, 29, 109$
Class number $512$ (GRH)
Class group $[2, 2, 128]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361429, -37721, 212671, -60735, 94594, -27887, 28250, -10096, 6639, -1930, 1142, -432, 147, -30, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 - 30*x^13 + 147*x^12 - 432*x^11 + 1142*x^10 - 1930*x^9 + 6639*x^8 - 10096*x^7 + 28250*x^6 - 27887*x^5 + 94594*x^4 - 60735*x^3 + 212671*x^2 - 37721*x + 361429)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 - 30*x^13 + 147*x^12 - 432*x^11 + 1142*x^10 - 1930*x^9 + 6639*x^8 - 10096*x^7 + 28250*x^6 - 27887*x^5 + 94594*x^4 - 60735*x^3 + 212671*x^2 - 37721*x + 361429, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} - 30 x^{13} + 147 x^{12} - 432 x^{11} + 1142 x^{10} - 1930 x^{9} + 6639 x^{8} - 10096 x^{7} + 28250 x^{6} - 27887 x^{5} + 94594 x^{4} - 60735 x^{3} + 212671 x^{2} - 37721 x + 361429 \)

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:  \(936758026692429703922265625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.50$
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:  $5, 13, 29, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{55} a^{14} - \frac{1}{11} a^{13} - \frac{2}{55} a^{12} + \frac{17}{55} a^{11} + \frac{12}{55} a^{10} + \frac{3}{55} a^{9} - \frac{1}{5} a^{8} + \frac{3}{55} a^{7} - \frac{24}{55} a^{6} - \frac{3}{55} a^{5} + \frac{23}{55} a^{4} + \frac{2}{55} a^{3} + \frac{6}{55} a^{2} + \frac{1}{5} a + \frac{13}{55}$, $\frac{1}{13598607245189786598688046868778025} a^{15} - \frac{60580377926356291202415367646443}{13598607245189786598688046868778025} a^{14} + \frac{425592469729340204820618612616356}{13598607245189786598688046868778025} a^{13} + \frac{774409671029782586909161964090983}{13598607245189786598688046868778025} a^{12} - \frac{2476569203433345907291565932472039}{13598607245189786598688046868778025} a^{11} + \frac{6608797668886802515974522505566466}{13598607245189786598688046868778025} a^{10} + \frac{1081115327072410069860417665723989}{2719721449037957319737609373755605} a^{9} - \frac{13055637723191250752246620345809}{2719721449037957319737609373755605} a^{8} + \frac{5709250577763296726571143407610359}{13598607245189786598688046868778025} a^{7} + \frac{1862880133421083501521192115150666}{13598607245189786598688046868778025} a^{6} - \frac{973985728065788234749750396028622}{13598607245189786598688046868778025} a^{5} + \frac{1085453483420213430034403023932727}{13598607245189786598688046868778025} a^{4} - \frac{904885871533835483869202522227758}{2719721449037957319737609373755605} a^{3} - \frac{1214855221119623972137549048329154}{2719721449037957319737609373755605} a^{2} - \frac{5507811696956709115783409105679294}{13598607245189786598688046868778025} a - \frac{1130465714562522423487374026581468}{13598607245189786598688046868778025}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{128}$, which has order $512$ (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:  \( 10968.6213178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.79025.1, 4.4.725.1, 4.0.2725.1, 8.0.30606503013125.1, 8.8.2576088125.1, 8.0.6244950625.5

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
109Data not computed