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Properties

Label 16.0.19494446360...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{8}$
Root discriminant $21.41$
Ramified primes $5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -56, 132, -254, 409, -836, 1478, -2111, 2108, 6, -327, 68, 10, 17, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 17*x^13 + 10*x^12 + 68*x^11 - 327*x^10 + 6*x^9 + 2108*x^8 - 2111*x^7 + 1478*x^6 - 836*x^5 + 409*x^4 - 254*x^3 + 132*x^2 - 56*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 17*x^13 + 10*x^12 + 68*x^11 - 327*x^10 + 6*x^9 + 2108*x^8 - 2111*x^7 + 1478*x^6 - 836*x^5 + 409*x^4 - 254*x^3 + 132*x^2 - 56*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 17 x^{13} + 10 x^{12} + 68 x^{11} - 327 x^{10} + 6 x^{9} + 2108 x^{8} - 2111 x^{7} + 1478 x^{6} - 836 x^{5} + 409 x^{4} - 254 x^{3} + 132 x^{2} - 56 x + 16 \)

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:  \(1949444636015869140625=5^{12}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.41$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{44592910} a^{13} + \frac{130842}{4459291} a^{12} - \frac{6779073}{22296455} a^{11} + \frac{3329785}{8918582} a^{10} + \frac{10707733}{22296455} a^{9} + \frac{1314787}{22296455} a^{8} - \frac{5486781}{44592910} a^{7} - \frac{6792592}{22296455} a^{6} + \frac{4913333}{22296455} a^{5} - \frac{11602271}{44592910} a^{4} + \frac{27107}{4459291} a^{3} + \frac{2425913}{22296455} a^{2} + \frac{2255155}{8918582} a + \frac{2266902}{22296455}$, $\frac{1}{89185820} a^{14} + \frac{1947607}{22296455} a^{12} + \frac{10790849}{89185820} a^{11} + \frac{1320174}{22296455} a^{10} + \frac{6031591}{22296455} a^{9} - \frac{8891647}{89185820} a^{8} - \frac{3042516}{22296455} a^{7} - \frac{4205852}{22296455} a^{6} - \frac{33375511}{89185820} a^{5} + \frac{7739371}{22296455} a^{4} - \frac{5286086}{22296455} a^{3} - \frac{2564891}{17837164} a^{2} - \frac{10313168}{22296455} a + \frac{909741}{22296455}$, $\frac{1}{891858200} a^{15} - \frac{1}{222964550} a^{13} + \frac{79492849}{891858200} a^{12} + \frac{62610347}{222964550} a^{11} + \frac{71830871}{222964550} a^{10} - \frac{286329959}{891858200} a^{9} + \frac{34359541}{111482275} a^{8} + \frac{94672601}{222964550} a^{7} - \frac{253350343}{891858200} a^{6} + \frac{50103169}{111482275} a^{5} + \frac{82950927}{222964550} a^{4} - \frac{4676147}{178371640} a^{3} - \frac{27148281}{222964550} a^{2} + \frac{60072881}{222964550} a - \frac{3933591}{111482275}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{764002}{22296455} a^{15} + \frac{3438009}{89185820} a^{14} + \frac{4202011}{22296455} a^{13} - \frac{9938089}{22296455} a^{12} - \frac{72962191}{89185820} a^{11} - \frac{61884162}{22296455} a^{10} + \frac{40492106}{4459291} a^{9} + \frac{160822421}{17837164} a^{8} - \frac{310650691}{4459291} a^{7} + \frac{40492106}{4459291} a^{6} - \frac{482467263}{89185820} a^{5} + \frac{53862141}{22296455} a^{4} - \frac{35908094}{22296455} a^{3} - \frac{1451591}{89185820} a^{2} - \frac{9168024}{22296455} a + \frac{3056008}{22296455} \) (order $10$)
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:  \( 39750.2370823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.5125.1 x2, 4.4.210125.1 x2, 4.0.8405.1 x2, 4.0.1025.1 x2, \(\Q(\zeta_{5})\), 4.0.210125.1, 8.8.44152515625.1, 8.0.1766100625.1, 8.0.44152515625.1, 8.0.26265625.1 x2, 8.0.44152515625.2 x2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$