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Properties

Label 16.0.42287613544...3041.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 41^{12}$
Root discriminant $39.96$
Ramified primes $37, 41$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100, 270, 1273, 2675, 6725, -2549, -2082, -772, -469, 268, 221, 115, 99, -5, 20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 20*x^14 - 5*x^13 + 99*x^12 + 115*x^11 + 221*x^10 + 268*x^9 - 469*x^8 - 772*x^7 - 2082*x^6 - 2549*x^5 + 6725*x^4 + 2675*x^3 + 1273*x^2 + 270*x + 100)
 
gp: K = bnfinit(x^16 - 2*x^15 + 20*x^14 - 5*x^13 + 99*x^12 + 115*x^11 + 221*x^10 + 268*x^9 - 469*x^8 - 772*x^7 - 2082*x^6 - 2549*x^5 + 6725*x^4 + 2675*x^3 + 1273*x^2 + 270*x + 100, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 20 x^{14} - 5 x^{13} + 99 x^{12} + 115 x^{11} + 221 x^{10} + 268 x^{9} - 469 x^{8} - 772 x^{7} - 2082 x^{6} - 2549 x^{5} + 6725 x^{4} + 2675 x^{3} + 1273 x^{2} + 270 x + 100 \)

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:  \(42287613544824591671753041=37^{4}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.96$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{80} a^{12} - \frac{1}{40} a^{11} + \frac{3}{40} a^{10} - \frac{1}{80} a^{9} - \frac{1}{40} a^{8} - \frac{1}{16} a^{7} - \frac{37}{80} a^{6} - \frac{29}{80} a^{5} + \frac{19}{80} a^{4} - \frac{33}{80} a^{3} - \frac{7}{80} a^{2} + \frac{11}{40} a + \frac{1}{4}$, $\frac{1}{160} a^{13} - \frac{1}{160} a^{12} + \frac{1}{40} a^{11} + \frac{1}{32} a^{10} + \frac{37}{160} a^{9} - \frac{7}{160} a^{8} - \frac{1}{80} a^{7} + \frac{27}{80} a^{6} - \frac{5}{16} a^{5} + \frac{33}{80} a^{4} + \frac{1}{4} a^{3} + \frac{11}{32} a^{2} - \frac{39}{80} a - \frac{3}{8}$, $\frac{1}{1920} a^{14} + \frac{1}{960} a^{13} + \frac{1}{384} a^{12} - \frac{31}{1920} a^{11} - \frac{11}{480} a^{10} + \frac{23}{96} a^{9} - \frac{191}{1920} a^{8} + \frac{29}{160} a^{7} - \frac{53}{160} a^{6} - \frac{7}{48} a^{5} + \frac{97}{960} a^{4} - \frac{37}{1920} a^{3} + \frac{33}{640} a^{2} + \frac{377}{960} a - \frac{29}{96}$, $\frac{1}{180735963202571946240} a^{15} - \frac{37953619763136719}{180735963202571946240} a^{14} - \frac{146799495597774303}{60245321067523982080} a^{13} + \frac{41020976373716989}{15061330266880995520} a^{12} - \frac{1316896072180020665}{36147192640514389248} a^{11} - \frac{921369606510198331}{7530665133440497760} a^{10} - \frac{14772712731325096489}{60245321067523982080} a^{9} - \frac{26145498417733141}{180735963202571946240} a^{8} + \frac{460323842725988765}{1506133026688099552} a^{7} + \frac{20600845449392682461}{45183990800642986560} a^{6} + \frac{2033697844809122071}{30122660533761991040} a^{5} + \frac{851503039000630951}{12049064213504796416} a^{4} + \frac{1826585148712553869}{4518399080064298656} a^{3} + \frac{72101005018202595151}{180735963202571946240} a^{2} + \frac{32205968057486402213}{90367981601285973120} a + \frac{1526146761952179485}{9036798160128597312}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 471474.859461 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.62197.1, 4.4.68921.1, 4.0.2550077.1, 8.0.158607139169.1 x2, 8.4.175753856917.1 x2, 8.0.6502892705929.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$