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Properties

Label 16.0.46871787785...9664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 23^{6}\cdot 37^{4}\cdot 24499031^{2}$
Root discriminant $536.33$
Ramified primes $2, 23, 37, 24499031$
Class number $21787464000$ (GRH)
Class group $[2, 2, 2, 2723433000]$ (GRH)
Galois group 16T1765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1270028532190841476, 0, 67056318133293792, 0, 1446242915789840, 0, 16792514721616, 0, 115758148212, 0, 488006872, 0, 1233350, 0, 1712, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1712*x^14 + 1233350*x^12 + 488006872*x^10 + 115758148212*x^8 + 16792514721616*x^6 + 1446242915789840*x^4 + 67056318133293792*x^2 + 1270028532190841476)
 
gp: K = bnfinit(x^16 + 1712*x^14 + 1233350*x^12 + 488006872*x^10 + 115758148212*x^8 + 16792514721616*x^6 + 1446242915789840*x^4 + 67056318133293792*x^2 + 1270028532190841476, 1)
 

Normalized defining polynomial

\( x^{16} + 1712 x^{14} + 1233350 x^{12} + 488006872 x^{10} + 115758148212 x^{8} + 16792514721616 x^{6} + 1446242915789840 x^{4} + 67056318133293792 x^{2} + 1270028532190841476 \)

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:  \(46871787785697452696878168826792882590449664=2^{48}\cdot 23^{6}\cdot 37^{4}\cdot 24499031^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $536.33$
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:  $2, 23, 37, 24499031$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{7374932726364029598563199270129651156487916928556} a^{14} - \frac{63457689815794013425175238260905646527106394031}{3687466363182014799281599635064825578243958464278} a^{12} - \frac{555596128209112045302163822035825844751011917791}{3687466363182014799281599635064825578243958464278} a^{10} + \frac{627483262069513862473047167401313307616455843833}{3687466363182014799281599635064825578243958464278} a^{8} + \frac{581897061429310749528311940746658898217946974539}{3687466363182014799281599635064825578243958464278} a^{6} + \frac{627250076595810715190492988811433320374526506857}{1843733181591007399640799817532412789121979232139} a^{4} - \frac{529661144837213304683093670395619921392201174873}{1843733181591007399640799817532412789121979232139} a^{2} + \frac{202403958646391967170058861182124648007}{3272060525295358752974848922786787823003}$, $\frac{1}{7374932726364029598563199270129651156487916928556} a^{15} - \frac{63457689815794013425175238260905646527106394031}{3687466363182014799281599635064825578243958464278} a^{13} - \frac{555596128209112045302163822035825844751011917791}{3687466363182014799281599635064825578243958464278} a^{11} + \frac{627483262069513862473047167401313307616455843833}{3687466363182014799281599635064825578243958464278} a^{9} + \frac{581897061429310749528311940746658898217946974539}{3687466363182014799281599635064825578243958464278} a^{7} + \frac{627250076595810715190492988811433320374526506857}{1843733181591007399640799817532412789121979232139} a^{5} - \frac{529661144837213304683093670395619921392201174873}{1843733181591007399640799817532412789121979232139} a^{3} + \frac{202403958646391967170058861182124648007}{3272060525295358752974848922786787823003} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2723433000}$, which has order $21787464000$ (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:  \( 109226.925145 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 74 conjugacy class representatives for t16n1765 are not computed
Character table for t16n1765 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.8.47461236736.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
24499031Data not computed