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Properties

Label 16.0.43980465111...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 5^{8}$
Root discriminant $19.51$
Ramified primes $2, 5$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -24, 164, -304, 288, -40, -190, 224, -64, -88, 124, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 124*x^12 - 88*x^11 - 64*x^10 + 224*x^9 - 190*x^8 - 40*x^7 + 288*x^6 - 304*x^5 + 164*x^4 - 24*x^3 + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 124*x^12 - 88*x^11 - 64*x^10 + 224*x^9 - 190*x^8 - 40*x^7 + 288*x^6 - 304*x^5 + 164*x^4 - 24*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 124 x^{12} - 88 x^{11} - 64 x^{10} + 224 x^{9} - 190 x^{8} - 40 x^{7} + 288 x^{6} - 304 x^{5} + 164 x^{4} - 24 x^{3} + 1 \)

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:  \(439804651110400000000=2^{50}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.51$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a - \frac{1}{16}$, $\frac{1}{48} a^{13} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{16} a^{9} - \frac{5}{12} a^{7} - \frac{5}{12} a^{6} - \frac{23}{48} a^{5} - \frac{1}{2} a^{4} - \frac{5}{24} a^{3} - \frac{11}{24} a^{2} + \frac{3}{16} a - \frac{1}{6}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{16} a^{10} + \frac{1}{48} a^{8} - \frac{5}{12} a^{7} - \frac{23}{48} a^{6} + \frac{11}{48} a^{4} - \frac{11}{24} a^{3} - \frac{5}{16} a^{2} + \frac{1}{3} a - \frac{5}{16}$, $\frac{1}{122928} a^{15} - \frac{971}{122928} a^{14} + \frac{85}{30732} a^{13} - \frac{751}{40976} a^{12} - \frac{7087}{122928} a^{11} + \frac{2189}{122928} a^{10} - \frac{23}{7683} a^{9} - \frac{6493}{122928} a^{8} - \frac{37247}{122928} a^{7} - \frac{1889}{40976} a^{6} + \frac{3871}{30732} a^{5} + \frac{42283}{122928} a^{4} - \frac{8025}{40976} a^{3} + \frac{53249}{122928} a^{2} - \frac{3194}{7683} a - \frac{1585}{122928}$

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:  \( -\frac{6005}{61464} a^{15} + \frac{42995}{61464} a^{14} - \frac{75841}{30732} a^{13} + \frac{158533}{30732} a^{12} - \frac{13818}{2561} a^{11} - \frac{30565}{15366} a^{10} + \frac{865315}{61464} a^{9} - \frac{344281}{20488} a^{8} - \frac{55603}{61464} a^{7} + \frac{1305877}{61464} a^{6} - \frac{194215}{7683} a^{5} + \frac{10923}{2561} a^{4} + \frac{319807}{30732} a^{3} - \frac{361835}{30732} a^{2} + \frac{64037}{61464} a + \frac{6371}{61464} \) (order $4$)
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:  \( 13929.2666566 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{10}) \), 4.2.25600.1 x2, \(\Q(i, \sqrt{10})\), 4.0.2560.2 x2, 8.4.10485760000.1, 8.0.10485760000.6, 8.0.2621440000.10

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$