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Properties

Label 16.0.65657364936...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $20.00$
Ramified primes $5, 29, 41$
Class number $2$
Class group $[2]$
Galois group $C_4.C_2^2:D_4$ (as 16T211)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 512, 1152, 160, 1232, 120, 1176, -232, 569, -17, 131, 5, 37, -10, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 12*x^14 - 10*x^13 + 37*x^12 + 5*x^11 + 131*x^10 - 17*x^9 + 569*x^8 - 232*x^7 + 1176*x^6 + 120*x^5 + 1232*x^4 + 160*x^3 + 1152*x^2 + 512*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 + 12*x^14 - 10*x^13 + 37*x^12 + 5*x^11 + 131*x^10 - 17*x^9 + 569*x^8 - 232*x^7 + 1176*x^6 + 120*x^5 + 1232*x^4 + 160*x^3 + 1152*x^2 + 512*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 12 x^{14} - 10 x^{13} + 37 x^{12} + 5 x^{11} + 131 x^{10} - 17 x^{9} + 569 x^{8} - 232 x^{7} + 1176 x^{6} + 120 x^{5} + 1232 x^{4} + 160 x^{3} + 1152 x^{2} + 512 x + 256 \)

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:  \(656573649364172265625=5^{8}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.00$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{7}{16} a^{8} - \frac{1}{16} a^{7} + \frac{5}{16} a^{6} + \frac{5}{16} a^{5} + \frac{7}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{7}{32} a^{9} + \frac{15}{32} a^{8} - \frac{11}{32} a^{7} + \frac{5}{32} a^{6} + \frac{7}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{7}{64} a^{10} + \frac{15}{64} a^{9} - \frac{11}{64} a^{8} - \frac{27}{64} a^{7} + \frac{7}{64} a^{6} - \frac{11}{32} a^{5} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{420334943881088} a^{15} + \frac{2510081700775}{420334943881088} a^{14} + \frac{3212602678087}{210167471940544} a^{13} - \frac{5766801856389}{210167471940544} a^{12} - \frac{24996946719687}{420334943881088} a^{11} - \frac{21284548007737}{420334943881088} a^{10} + \frac{13136392629957}{420334943881088} a^{9} - \frac{9210923649355}{420334943881088} a^{8} - \frac{76942524685145}{420334943881088} a^{7} + \frac{74040826618541}{210167471940544} a^{6} - \frac{12448807481465}{26270933992568} a^{5} - \frac{5764539234757}{13135466996284} a^{4} - \frac{3929292281191}{26270933992568} a^{3} + \frac{634329953023}{3283866749071} a^{2} + \frac{1654989429781}{6567733498142} a + \frac{274161543616}{3283866749071}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1493.70001813 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.29725.2, 4.0.1025.1, 8.4.25623693125.1, 8.4.15243125.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$