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Properties

Label 16.0.72646052899...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{14}$
Root discriminant $63.65$
Ramified primes $5, 29$
Class number $96$ (GRH)
Class group $[4, 24]$ (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![576, 1890, 10717, 17365, 43715, 24430, 30272, 9955, 7554, 1220, -312, 205, -130, -20, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 23*x^14 - 20*x^13 - 130*x^12 + 205*x^11 - 312*x^10 + 1220*x^9 + 7554*x^8 + 9955*x^7 + 30272*x^6 + 24430*x^5 + 43715*x^4 + 17365*x^3 + 10717*x^2 + 1890*x + 576)
 
gp: K = bnfinit(x^16 - 5*x^15 + 23*x^14 - 20*x^13 - 130*x^12 + 205*x^11 - 312*x^10 + 1220*x^9 + 7554*x^8 + 9955*x^7 + 30272*x^6 + 24430*x^5 + 43715*x^4 + 17365*x^3 + 10717*x^2 + 1890*x + 576, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 23 x^{14} - 20 x^{13} - 130 x^{12} + 205 x^{11} - 312 x^{10} + 1220 x^{9} + 7554 x^{8} + 9955 x^{7} + 30272 x^{6} + 24430 x^{5} + 43715 x^{4} + 17365 x^{3} + 10717 x^{2} + 1890 x + 576 \)

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:  \(72646052899365103388916015625=5^{12}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.65$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{180} a^{12} + \frac{1}{180} a^{11} + \frac{1}{20} a^{10} - \frac{7}{45} a^{9} + \frac{1}{180} a^{8} + \frac{37}{90} a^{7} - \frac{83}{180} a^{6} - \frac{22}{45} a^{5} - \frac{9}{20} a^{4} + \frac{49}{180} a^{3} - \frac{41}{180} a^{2} + \frac{13}{30} a - \frac{1}{5}$, $\frac{1}{360} a^{13} + \frac{1}{45} a^{11} + \frac{23}{360} a^{10} - \frac{31}{360} a^{9} - \frac{47}{360} a^{8} - \frac{37}{360} a^{7} + \frac{11}{72} a^{6} - \frac{53}{360} a^{5} + \frac{1}{36} a^{4} - \frac{5}{12} a^{3} - \frac{61}{360} a^{2} + \frac{11}{60} a - \frac{2}{5}$, $\frac{1}{12960} a^{14} + \frac{7}{12960} a^{13} - \frac{1}{405} a^{12} - \frac{1}{160} a^{11} - \frac{11}{1296} a^{10} - \frac{7}{405} a^{9} - \frac{983}{6480} a^{8} - \frac{941}{3240} a^{7} + \frac{703}{3240} a^{6} + \frac{1333}{4320} a^{5} - \frac{79}{162} a^{4} + \frac{4009}{12960} a^{3} + \frac{6079}{12960} a^{2} + \frac{19}{80} a + \frac{14}{45}$, $\frac{1}{11192228776488913390080} a^{15} - \frac{29225815703143217}{2798057194122228347520} a^{14} - \frac{822461867791399277}{11192228776488913390080} a^{13} - \frac{2395189701036822571}{3730742925496304463360} a^{12} - \frac{36255727336362599747}{11192228776488913390080} a^{11} + \frac{399810855927514603589}{5596114388244456695040} a^{10} - \frac{34422076914418158955}{1119222877648891339008} a^{9} - \frac{523053833132040057013}{5596114388244456695040} a^{8} + \frac{84824512095470046803}{1399028597061114173760} a^{7} - \frac{95831268895355660089}{248716195033086964224} a^{6} + \frac{323201264388776489851}{11192228776488913390080} a^{5} - \frac{1274925307991819425559}{11192228776488913390080} a^{4} + \frac{1304574461998061208379}{2798057194122228347520} a^{3} + \frac{215757853230871149295}{746148585099260892672} a^{2} + \frac{2890973628893121509}{14460243897272497920} a + \frac{1521287787651349949}{6476984245653306360}$

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

Class group and class number

$C_{4}\times C_{24}$, which has order $96$ (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:  \( 33790831.4592 \) (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

\(\Q(\sqrt{145}) \), 4.0.609725.1, 4.4.3048625.1, 4.0.105125.1, 8.0.53905863465625.2, 8.0.53905863465625.1, 8.0.9294114390625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$