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Properties

Label 16.0.17669691567...3761.7
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{12}$
Root discriminant $77.71$
Ramified primes $23, 41$
Class number $48$ (GRH)
Class group $[2, 24]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![133225, 136510, 136103, -395560, 128687, 148276, -95415, -46316, 37978, 11648, -6837, -1744, 681, 130, -38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 38*x^14 + 130*x^13 + 681*x^12 - 1744*x^11 - 6837*x^10 + 11648*x^9 + 37978*x^8 - 46316*x^7 - 95415*x^6 + 148276*x^5 + 128687*x^4 - 395560*x^3 + 136103*x^2 + 136510*x + 133225)
 
gp: K = bnfinit(x^16 - 4*x^15 - 38*x^14 + 130*x^13 + 681*x^12 - 1744*x^11 - 6837*x^10 + 11648*x^9 + 37978*x^8 - 46316*x^7 - 95415*x^6 + 148276*x^5 + 128687*x^4 - 395560*x^3 + 136103*x^2 + 136510*x + 133225, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 38 x^{14} + 130 x^{13} + 681 x^{12} - 1744 x^{11} - 6837 x^{10} + 11648 x^{9} + 37978 x^{8} - 46316 x^{7} - 95415 x^{6} + 148276 x^{5} + 128687 x^{4} - 395560 x^{3} + 136103 x^{2} + 136510 x + 133225 \)

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:  \(1766969156799962667099298073761=23^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.71$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{20} a^{7} + \frac{1}{20} a^{6} + \frac{2}{5} a^{5} - \frac{9}{20} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{7}{20} a + \frac{1}{4}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{5} + \frac{3}{20} a^{4} + \frac{3}{10} a^{3} - \frac{1}{4} a^{2} - \frac{1}{20} a - \frac{1}{4}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} + \frac{1}{20} a^{9} + \frac{3}{40} a^{8} - \frac{3}{40} a^{6} + \frac{1}{5} a^{5} - \frac{7}{20} a^{4} - \frac{1}{4} a^{3} + \frac{7}{20} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{11} - \frac{3}{40} a^{9} + \frac{1}{10} a^{8} + \frac{1}{8} a^{7} - \frac{1}{10} a^{6} + \frac{1}{4} a^{5} + \frac{9}{20} a^{4} - \frac{1}{5} a^{3} - \frac{9}{20} a^{2} - \frac{9}{40} a - \frac{1}{4}$, $\frac{1}{80} a^{14} - \frac{1}{80} a^{13} + \frac{1}{80} a^{11} + \frac{1}{80} a^{9} + \frac{3}{40} a^{8} + \frac{3}{16} a^{7} - \frac{3}{16} a^{6} - \frac{1}{40} a^{5} - \frac{1}{5} a^{4} - \frac{3}{40} a^{3} - \frac{11}{80} a^{2} - \frac{3}{80} a - \frac{5}{16}$, $\frac{1}{18736023267731286510490632925040} a^{15} + \frac{38369984867081185501947158063}{9368011633865643255245316462520} a^{14} - \frac{35673939905935746327009419831}{3747204653546257302098126585008} a^{13} - \frac{206057807483278393998956240017}{18736023267731286510490632925040} a^{12} + \frac{343361865994812797780314528819}{18736023267731286510490632925040} a^{11} - \frac{189744256320074633481442776233}{18736023267731286510490632925040} a^{10} - \frac{478459232786882225226587019339}{18736023267731286510490632925040} a^{9} + \frac{1254981294383699387265170511283}{18736023267731286510490632925040} a^{8} - \frac{87845604153298187569295700675}{1873602326773128651049063292504} a^{7} + \frac{642700573905821880605347992967}{18736023267731286510490632925040} a^{6} + \frac{2986503257315345487304215419549}{9368011633865643255245316462520} a^{5} + \frac{746722450292667807121414112645}{1873602326773128651049063292504} a^{4} + \frac{6289668777904199002804467029983}{18736023267731286510490632925040} a^{3} - \frac{1213551743482266191832554585939}{4684005816932821627622658231260} a^{2} + \frac{2807741666315590920234669813469}{9368011633865643255245316462520} a + \frac{25657126212418835274102671883}{51331570596524072631481186096}$

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

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (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:  \( 15364202.9179 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

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

\(\Q(\sqrt{41}) \), 4.0.36459209.2, 4.2.38663.1, 4.2.1585183.1, 8.0.61287930329.1, 8.4.32421315144041.1, 8.0.1329273920905681.4

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$23.4.2.1$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}$
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}$
$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}$