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Properties

Label 16.0.19539757911...7281.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 53^{8}$
Root discriminant $43.97$
Ramified primes $11, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42529, 133159, 215491, 108305, 27154, -39988, 13730, -10479, 9324, -2924, 2998, -417, 373, -21, 27, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 27*x^14 - 21*x^13 + 373*x^12 - 417*x^11 + 2998*x^10 - 2924*x^9 + 9324*x^8 - 10479*x^7 + 13730*x^6 - 39988*x^5 + 27154*x^4 + 108305*x^3 + 215491*x^2 + 133159*x + 42529)
 
gp: K = bnfinit(x^16 - x^15 + 27*x^14 - 21*x^13 + 373*x^12 - 417*x^11 + 2998*x^10 - 2924*x^9 + 9324*x^8 - 10479*x^7 + 13730*x^6 - 39988*x^5 + 27154*x^4 + 108305*x^3 + 215491*x^2 + 133159*x + 42529, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 27 x^{14} - 21 x^{13} + 373 x^{12} - 417 x^{11} + 2998 x^{10} - 2924 x^{9} + 9324 x^{8} - 10479 x^{7} + 13730 x^{6} - 39988 x^{5} + 27154 x^{4} + 108305 x^{3} + 215491 x^{2} + 133159 x + 42529 \)

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:  \(195397579112879711966327281=11^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.97$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{2} - \frac{3}{7}$, $\frac{1}{21} a^{11} + \frac{1}{21} a^{10} + \frac{2}{21} a^{9} + \frac{1}{3} a^{8} - \frac{4}{21} a^{7} - \frac{10}{21} a^{6} + \frac{8}{21} a^{5} + \frac{2}{21} a^{4} + \frac{4}{21} a^{3} + \frac{10}{21} a^{2} - \frac{1}{7} a + \frac{10}{21}$, $\frac{1}{21} a^{12} + \frac{1}{21} a^{10} + \frac{5}{21} a^{9} + \frac{10}{21} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{21} a^{4} + \frac{2}{7} a^{3} + \frac{8}{21} a^{2} - \frac{8}{21} a - \frac{10}{21}$, $\frac{1}{147} a^{13} + \frac{1}{147} a^{12} - \frac{2}{147} a^{11} + \frac{3}{49} a^{10} + \frac{1}{7} a^{9} - \frac{2}{21} a^{8} + \frac{16}{49} a^{7} - \frac{25}{147} a^{5} - \frac{31}{147} a^{4} + \frac{2}{147} a^{3} - \frac{23}{49} a^{2} - \frac{3}{49} a - \frac{37}{147}$, $\frac{1}{735} a^{14} - \frac{2}{735} a^{13} - \frac{4}{245} a^{12} + \frac{8}{735} a^{11} + \frac{22}{735} a^{10} - \frac{2}{35} a^{9} - \frac{302}{735} a^{8} - \frac{40}{147} a^{7} + \frac{22}{245} a^{6} + \frac{23}{49} a^{5} - \frac{311}{735} a^{4} - \frac{292}{735} a^{3} - \frac{116}{245} a^{2} - \frac{16}{147} a + \frac{93}{245}$, $\frac{1}{547112497626868050612226402387995} a^{15} + \frac{57889413143716767340746605366}{547112497626868050612226402387995} a^{14} + \frac{324520824617472226911833223319}{182370832542289350204075467462665} a^{13} - \frac{10893422480077239132455249016778}{547112497626868050612226402387995} a^{12} - \frac{1064578831996447007403722196754}{547112497626868050612226402387995} a^{11} + \frac{28044225237913391642891759139494}{547112497626868050612226402387995} a^{10} - \frac{57955929624304280885339514065496}{182370832542289350204075467462665} a^{9} + \frac{26710704236528853130787025978224}{547112497626868050612226402387995} a^{8} + \frac{22753881587160875185908421160836}{547112497626868050612226402387995} a^{7} - \frac{4491709914639404075501010061534}{182370832542289350204075467462665} a^{6} + \frac{236602938755463516675828791987264}{547112497626868050612226402387995} a^{5} + \frac{887443834391201099521534781306}{5210595215493981434402156213219} a^{4} - \frac{3467613300409697655694246656498}{182370832542289350204075467462665} a^{3} + \frac{29200344018977863702265675483842}{182370832542289350204075467462665} a^{2} - \frac{268406072657092631917700122379656}{547112497626868050612226402387995} a - \frac{80496982015553898103586279271141}{182370832542289350204075467462665}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 4622143.17972 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.6413.1, 8.0.2179708157.1, 8.0.263744686997.1, 8.0.13978468410841.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$