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Properties

Label 16.0.11846085363...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 29^{6}$
Root discriminant $42.62$
Ramified primes $5, 13, 29$
Class number $56$ (GRH)
Class group $[2, 28]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5836, -9660, 23275, -33452, 48622, -58860, 63469, -48408, 25319, -9468, 2729, -704, 159, -24, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 24*x^13 + 159*x^12 - 704*x^11 + 2729*x^10 - 9468*x^9 + 25319*x^8 - 48408*x^7 + 63469*x^6 - 58860*x^5 + 48622*x^4 - 33452*x^3 + 23275*x^2 - 9660*x + 5836)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 24*x^13 + 159*x^12 - 704*x^11 + 2729*x^10 - 9468*x^9 + 25319*x^8 - 48408*x^7 + 63469*x^6 - 58860*x^5 + 48622*x^4 - 33452*x^3 + 23275*x^2 - 9660*x + 5836, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 24 x^{13} + 159 x^{12} - 704 x^{11} + 2729 x^{10} - 9468 x^{9} + 25319 x^{8} - 48408 x^{7} + 63469 x^{6} - 58860 x^{5} + 48622 x^{4} - 33452 x^{3} + 23275 x^{2} - 9660 x + 5836 \)

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:  \(118460853639390732666015625=5^{12}\cdot 13^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.62$
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:  $5, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a + \frac{1}{8}$, $\frac{1}{320} a^{12} + \frac{1}{80} a^{11} + \frac{9}{320} a^{10} - \frac{1}{20} a^{9} - \frac{3}{320} a^{8} + \frac{1}{16} a^{7} + \frac{7}{64} a^{6} - \frac{1}{40} a^{5} - \frac{17}{320} a^{4} - \frac{33}{80} a^{3} - \frac{17}{64} a^{2} + \frac{19}{40} a + \frac{29}{80}$, $\frac{1}{320} a^{13} - \frac{7}{320} a^{11} - \frac{3}{80} a^{10} - \frac{19}{320} a^{9} + \frac{1}{10} a^{8} - \frac{1}{64} a^{7} - \frac{17}{80} a^{6} + \frac{3}{64} a^{5} + \frac{7}{40} a^{4} + \frac{43}{320} a^{3} - \frac{37}{80} a^{2} - \frac{13}{80} a - \frac{1}{5}$, $\frac{1}{37120} a^{14} + \frac{9}{37120} a^{13} - \frac{7}{9280} a^{12} - \frac{819}{37120} a^{11} + \frac{1}{9280} a^{10} - \frac{2263}{37120} a^{9} - \frac{307}{18560} a^{8} - \frac{2393}{37120} a^{7} + \frac{1257}{9280} a^{6} + \frac{2459}{37120} a^{5} + \frac{343}{4640} a^{4} - \frac{8449}{37120} a^{3} + \frac{8161}{37120} a^{2} + \frac{1167}{4640} a + \frac{1437}{9280}$, $\frac{1}{1307389428137626359040} a^{15} - \frac{2853591266604631}{653694714068813179520} a^{14} - \frac{238437393794744723}{1307389428137626359040} a^{13} - \frac{2020831171932912831}{1307389428137626359040} a^{12} - \frac{2029744724031969715}{261477885627525271808} a^{11} + \frac{10633161722785374585}{261477885627525271808} a^{10} + \frac{38259720188009399763}{1307389428137626359040} a^{9} - \frac{65862531833790959231}{1307389428137626359040} a^{8} + \frac{40951823730573710971}{1307389428137626359040} a^{7} - \frac{1130915177680693991}{18413935607572202240} a^{6} + \frac{16436787026083533639}{261477885627525271808} a^{5} - \frac{37962575962386474169}{1307389428137626359040} a^{4} + \frac{40258853544892135937}{81711839258601647440} a^{3} - \frac{95086368172421990767}{1307389428137626359040} a^{2} + \frac{162027639107941376831}{326847357034406589760} a + \frac{52272239769375005469}{326847357034406589760}$

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

Class group and class number

$C_{2}\times C_{28}$, which has order $56$ (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:  \( 336977.682359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), 4.0.3625.1, 4.0.612625.1, 4.0.122525.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.122525.2, 8.0.15012375625.1, 8.0.375309390625.1, 8.8.375309390625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.6.1$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$