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Properties

Label 16.0.12260383203...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 89^{4}$
Root discriminant $36.99$
Ramified primes $5, 29, 89$
Class number $182$ (GRH)
Class group $[182]$ (GRH)
Galois group $C_2\wr C_2^2$ (as 16T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28775, -20900, 68140, -45560, 63804, -37970, 32508, -16676, 10025, -4678, 2046, -824, 309, -92, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 92*x^13 + 309*x^12 - 824*x^11 + 2046*x^10 - 4678*x^9 + 10025*x^8 - 16676*x^7 + 32508*x^6 - 37970*x^5 + 63804*x^4 - 45560*x^3 + 68140*x^2 - 20900*x + 28775)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 92*x^13 + 309*x^12 - 824*x^11 + 2046*x^10 - 4678*x^9 + 10025*x^8 - 16676*x^7 + 32508*x^6 - 37970*x^5 + 63804*x^4 - 45560*x^3 + 68140*x^2 - 20900*x + 28775, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 92 x^{13} + 309 x^{12} - 824 x^{11} + 2046 x^{10} - 4678 x^{9} + 10025 x^{8} - 16676 x^{7} + 32508 x^{6} - 37970 x^{5} + 63804 x^{4} - 45560 x^{3} + 68140 x^{2} - 20900 x + 28775 \)

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:  \(12260383203665853750390625=5^{8}\cdot 29^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.99$
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:  $5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{60} a^{12} + \frac{13}{60} a^{11} - \frac{1}{4} a^{10} - \frac{3}{20} a^{9} - \frac{7}{60} a^{8} + \frac{7}{30} a^{7} + \frac{1}{4} a^{6} - \frac{3}{20} a^{5} + \frac{2}{5} a^{4} - \frac{1}{15} a^{3} + \frac{13}{30} a^{2} - \frac{1}{12} a + \frac{1}{12}$, $\frac{1}{60} a^{13} - \frac{1}{15} a^{11} + \frac{1}{10} a^{10} - \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{13}{60} a^{7} - \frac{2}{5} a^{6} + \frac{7}{20} a^{5} - \frac{4}{15} a^{4} + \frac{3}{10} a^{3} + \frac{17}{60} a^{2} + \frac{1}{6} a - \frac{1}{12}$, $\frac{1}{60} a^{14} - \frac{1}{30} a^{11} - \frac{1}{6} a^{10} + \frac{3}{20} a^{9} - \frac{1}{4} a^{8} - \frac{7}{15} a^{7} + \frac{7}{20} a^{6} + \frac{2}{15} a^{5} - \frac{1}{10} a^{4} + \frac{1}{60} a^{3} - \frac{1}{10} a^{2} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{134965226166481167216044100} a^{15} + \frac{217828663473944426695843}{67482613083240583608022050} a^{14} + \frac{294782835202216797361889}{134965226166481167216044100} a^{13} + \frac{290941782396528698000621}{134965226166481167216044100} a^{12} + \frac{2812453661529806886970427}{44988408722160389072014700} a^{11} + \frac{2445545178188159030699437}{33741306541620291804011025} a^{10} + \frac{1821070055972099050472}{75147676039243411590225} a^{9} - \frac{14506526573235709006001167}{67482613083240583608022050} a^{8} + \frac{25221433847929836299999071}{67482613083240583608022050} a^{7} + \frac{3869079461864108134010423}{134965226166481167216044100} a^{6} + \frac{4710557160974199204787006}{33741306541620291804011025} a^{5} - \frac{64556804414732720100882097}{134965226166481167216044100} a^{4} + \frac{3533134898830479027383329}{13496522616648116721604410} a^{3} + \frac{586716116731411915089103}{6748261308324058360802205} a^{2} - \frac{151955747432971839861749}{1799536348886415562880588} a + \frac{536158212625193451906665}{1349652261664811672160441}$

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

Class group and class number

$C_{182}$, which has order $182$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T150):

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

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.0.140059320025.1, 8.8.442050625.1, 8.0.3501483000625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$