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Properties

Label 16.0.24427448601...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{6}\cdot 41^{6}\cdot 97^{4}$
Root discriminant $122.11$
Ramified primes $5, 29, 41, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47938335701, -14310734897, 5609396159, -919387662, 179699696, -3867184, -2112483, 3596219, -841908, 283427, -19522, 2057, 2231, -236, 96, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 96*x^14 - 236*x^13 + 2231*x^12 + 2057*x^11 - 19522*x^10 + 283427*x^9 - 841908*x^8 + 3596219*x^7 - 2112483*x^6 - 3867184*x^5 + 179699696*x^4 - 919387662*x^3 + 5609396159*x^2 - 14310734897*x + 47938335701)
 
gp: K = bnfinit(x^16 - 4*x^15 + 96*x^14 - 236*x^13 + 2231*x^12 + 2057*x^11 - 19522*x^10 + 283427*x^9 - 841908*x^8 + 3596219*x^7 - 2112483*x^6 - 3867184*x^5 + 179699696*x^4 - 919387662*x^3 + 5609396159*x^2 - 14310734897*x + 47938335701, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 96 x^{14} - 236 x^{13} + 2231 x^{12} + 2057 x^{11} - 19522 x^{10} + 283427 x^{9} - 841908 x^{8} + 3596219 x^{7} - 2112483 x^{6} - 3867184 x^{5} + 179699696 x^{4} - 919387662 x^{3} + 5609396159 x^{2} - 14310734897 x + 47938335701 \)

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:  \(2442744860101307576054633212890625=5^{10}\cdot 29^{6}\cdot 41^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.11$
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:  $5, 29, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{75} a^{14} + \frac{2}{75} a^{13} - \frac{4}{75} a^{12} - \frac{34}{75} a^{11} + \frac{1}{5} a^{9} - \frac{7}{75} a^{8} + \frac{1}{15} a^{7} + \frac{31}{75} a^{6} - \frac{2}{5} a^{5} - \frac{2}{15} a^{4} - \frac{34}{75} a^{3} + \frac{4}{25} a^{2} - \frac{32}{75} a + \frac{23}{75}$, $\frac{1}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{15} - \frac{3311997217667048305265831475009766124528005396127396880672489747399}{600872083106710051749285884582043314514406544235509339796841964203325} a^{14} - \frac{32698799769576139896585277271657918126263528277626364320406581584047}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{13} - \frac{153493004908374568675215743236623278904732980337057695256812584609268}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{12} + \frac{305362229224127096881817951114594461272973255869717943294245113008841}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{11} + \frac{56742589474859354361263519880847128024653215321546521009240467040309}{120174416621342010349857176916408662902881308847101867959368392840665} a^{10} + \frac{312619595945633943026786639238038323149962814124563244224186479637083}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{9} + \frac{401759206857365142026055551627677883267816921528349296066501377252783}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{8} - \frac{141472651918270023565254173180641949770177558972583168348528100268763}{600872083106710051749285884582043314514406544235509339796841964203325} a^{7} + \frac{857531445422549418911708204666988738010658656401663421560000485314846}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{6} - \frac{118903833685402927711649514199180296413216017709955133708370806630943}{360523249864026031049571530749225988708643926541305603878105178521995} a^{5} + \frac{357724332022822626515671149784212764034247594857022585160534795197246}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{4} - \frac{130182490417315842938393743397177023374964428579509871451012885111282}{1802616249320130155247857653746129943543219632706528019390525892609975} a^{3} + \frac{19728637634538724416394334922018746383321022137858074493990880489519}{360523249864026031049571530749225988708643926541305603878105178521995} a^{2} - \frac{159225150758095081348563742243968654975587895028715962642843325512078}{600872083106710051749285884582043314514406544235509339796841964203325} a + \frac{308877392704396031240073171050514211595580930219298873215652856679293}{1802616249320130155247857653746129943543219632706528019390525892609975}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1589761353.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.1, 4.4.725.1, 4.0.1025.1, 8.0.883575625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$