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Properties

Label 16.0.15155900128...8816.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 11^{10}$
Root discriminant $57.71$
Ramified primes $2, 3, 11$
Class number $128$ (GRH)
Class group $[2, 4, 4, 4]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1185921, 0, 490050, 0, 102006, 0, 13770, 0, -54, 0, -594, 0, -42, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 42*x^12 - 594*x^10 - 54*x^8 + 13770*x^6 + 102006*x^4 + 490050*x^2 + 1185921)
 
gp: K = bnfinit(x^16 + 6*x^14 - 42*x^12 - 594*x^10 - 54*x^8 + 13770*x^6 + 102006*x^4 + 490050*x^2 + 1185921, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 42 x^{12} - 594 x^{10} - 54 x^{8} + 13770 x^{6} + 102006 x^{4} + 490050 x^{2} + 1185921 \)

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:  \(15155900128400657740323618816=2^{40}\cdot 3^{12}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.71$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{9} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{54} a^{8} + \frac{1}{9} a^{4} + \frac{1}{6}$, $\frac{1}{54} a^{9} + \frac{1}{9} a^{5} + \frac{1}{6} a$, $\frac{1}{54} a^{10} - \frac{1}{6} a^{2}$, $\frac{1}{54} a^{11} - \frac{1}{6} a^{3}$, $\frac{1}{5832} a^{12} - \frac{1}{108} a^{10} + \frac{1}{648} a^{8} + \frac{5}{72} a^{4} + \frac{7}{36} a^{2} + \frac{79}{216}$, $\frac{1}{128304} a^{13} - \frac{1}{11664} a^{12} - \frac{17}{2376} a^{11} - \frac{1}{216} a^{10} + \frac{109}{14256} a^{9} - \frac{1}{1296} a^{8} - \frac{1}{18} a^{7} - \frac{19}{1584} a^{5} - \frac{5}{144} a^{4} + \frac{283}{792} a^{3} + \frac{35}{72} a^{2} - \frac{677}{4752} a - \frac{79}{432}$, $\frac{1}{108835792560} a^{14} - \frac{56797}{7255719504} a^{12} - \frac{1}{108} a^{11} + \frac{80781007}{12092865840} a^{10} - \frac{1}{108} a^{9} - \frac{2680141}{366450480} a^{8} - \frac{1}{18} a^{7} - \frac{64113827}{1343651760} a^{6} - \frac{1}{18} a^{5} + \frac{50228117}{1343651760} a^{4} + \frac{5}{12} a^{3} + \frac{57558517}{4030955280} a^{2} + \frac{5}{12} a - \frac{4409249}{11104560}$, $\frac{1}{1197193718160} a^{15} - \frac{56797}{79812914544} a^{13} + \frac{976548847}{133021524240} a^{11} - \frac{1}{108} a^{10} + \frac{10892099}{4030955280} a^{9} - \frac{1}{108} a^{8} - \frac{64113827}{14780169360} a^{7} - \frac{1}{18} a^{6} - \frac{99066523}{14780169360} a^{5} - \frac{1}{18} a^{4} + \frac{17525031397}{44340508080} a^{3} + \frac{5}{12} a^{2} - \frac{56230529}{122150160} a + \frac{5}{12}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}$, which has order $128$ (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:  \( 790696.403358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{33}) \), 4.4.13824.1, 4.4.1672704.1, \(\Q(\sqrt{3}, \sqrt{11})\), 8.8.2797938671616.2, 8.0.427462852608.2, 8.0.3847165673472.16

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.20.88$x^{8} + 6 x^{6} + 4 x^{5} + 12 x^{2} + 6$$8$$1$$20$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
2.8.20.86$x^{8} + 16 x^{5} + 20$$8$$1$$20$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$