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Properties

Label 16.16.7387029288...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{58}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $47.78$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9071, -55128, -98608, -16600, 116708, 80248, -45776, -50920, 7794, 15080, -688, -2456, 92, 216, -16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 16*x^14 + 216*x^13 + 92*x^12 - 2456*x^11 - 688*x^10 + 15080*x^9 + 7794*x^8 - 50920*x^7 - 45776*x^6 + 80248*x^5 + 116708*x^4 - 16600*x^3 - 98608*x^2 - 55128*x - 9071)
 
gp: K = bnfinit(x^16 - 8*x^15 - 16*x^14 + 216*x^13 + 92*x^12 - 2456*x^11 - 688*x^10 + 15080*x^9 + 7794*x^8 - 50920*x^7 - 45776*x^6 + 80248*x^5 + 116708*x^4 - 16600*x^3 - 98608*x^2 - 55128*x - 9071, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 16 x^{14} + 216 x^{13} + 92 x^{12} - 2456 x^{11} - 688 x^{10} + 15080 x^{9} + 7794 x^{8} - 50920 x^{7} - 45776 x^{6} + 80248 x^{5} + 116708 x^{4} - 16600 x^{3} - 98608 x^{2} - 55128 x - 9071 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(738702928879445606400000000=2^{58}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.78$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{2} a^{6} + \frac{5}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{7}{16} a - \frac{5}{16}$, $\frac{1}{592} a^{14} - \frac{11}{592} a^{13} + \frac{11}{296} a^{12} - \frac{1}{37} a^{11} - \frac{9}{592} a^{10} + \frac{7}{592} a^{9} - \frac{7}{296} a^{8} - \frac{17}{37} a^{7} - \frac{155}{592} a^{6} - \frac{15}{592} a^{5} + \frac{131}{296} a^{4} - \frac{25}{74} a^{3} + \frac{255}{592} a^{2} + \frac{135}{592} a - \frac{103}{296}$, $\frac{1}{51986546896} a^{15} + \frac{41335185}{51986546896} a^{14} + \frac{309404513}{12996636724} a^{13} + \frac{73021821}{3249159181} a^{12} - \frac{1490353695}{51986546896} a^{11} - \frac{1832523529}{51986546896} a^{10} + \frac{796293943}{25993273448} a^{9} + \frac{771561781}{25993273448} a^{8} + \frac{1090778457}{51986546896} a^{7} + \frac{2490435789}{51986546896} a^{6} - \frac{194657839}{3249159181} a^{5} - \frac{2767988639}{12996636724} a^{4} - \frac{96866503}{1405041808} a^{3} - \frac{25639708041}{51986546896} a^{2} + \frac{6567382145}{25993273448} a + \frac{11391906715}{25993273448}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 303639501.725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.92160.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.92160.2, 8.8.33973862400.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$