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Properties

Label 16.16.2090117913...4096.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{58}\cdot 3^{14}\cdot 11^{8}\cdot 29^{4}$
Root discriminant $248.32$
Ramified primes $2, 3, 11, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11276922920769, 0, -3136585001616, 0, 339988792572, 0, -18985739256, 0, 599045106, 0, -10886304, 0, 109956, 0, -552, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 552*x^14 + 109956*x^12 - 10886304*x^10 + 599045106*x^8 - 18985739256*x^6 + 339988792572*x^4 - 3136585001616*x^2 + 11276922920769)
 
gp: K = bnfinit(x^16 - 552*x^14 + 109956*x^12 - 10886304*x^10 + 599045106*x^8 - 18985739256*x^6 + 339988792572*x^4 - 3136585001616*x^2 + 11276922920769, 1)
 

Normalized defining polynomial

\( x^{16} - 552 x^{14} + 109956 x^{12} - 10886304 x^{10} + 599045106 x^{8} - 18985739256 x^{6} + 339988792572 x^{4} - 3136585001616 x^{2} + 11276922920769 \)

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:  \(209011791362878677931936734396700164096=2^{58}\cdot 3^{14}\cdot 11^{8}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.32$
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:  $2, 3, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} - \frac{2}{11} a^{4}$, $\frac{1}{11} a^{7} - \frac{2}{11} a^{5}$, $\frac{1}{2904} a^{8} - \frac{1}{121} a^{6} + \frac{3}{44} a^{4} - \frac{4}{11} a^{2} - \frac{3}{8}$, $\frac{1}{2904} a^{9} - \frac{1}{121} a^{7} + \frac{3}{44} a^{5} - \frac{4}{11} a^{3} - \frac{3}{8} a$, $\frac{1}{84216} a^{10} - \frac{1}{84216} a^{8} - \frac{147}{14036} a^{6} - \frac{195}{1276} a^{4} - \frac{857}{2552} a^{2} - \frac{1}{8}$, $\frac{1}{84216} a^{11} - \frac{1}{84216} a^{9} - \frac{147}{14036} a^{7} - \frac{195}{1276} a^{5} - \frac{857}{2552} a^{3} - \frac{1}{8} a$, $\frac{1}{617892792} a^{12} - \frac{595}{308946396} a^{10} + \frac{626}{7021509} a^{8} - \frac{31113}{4681006} a^{6} + \frac{292801}{1702184} a^{4} + \frac{14217}{29348} a^{2} + \frac{15}{92}$, $\frac{1}{617892792} a^{13} - \frac{595}{308946396} a^{11} + \frac{626}{7021509} a^{9} - \frac{31113}{4681006} a^{7} + \frac{292801}{1702184} a^{5} + \frac{14217}{29348} a^{3} + \frac{15}{92} a$, $\frac{1}{13014519901143013551672} a^{14} + \frac{308296431175}{2169086650190502258612} a^{12} + \frac{925352336701949}{197189695471863841692} a^{10} - \frac{52723422822067171}{394379390943727683384} a^{8} + \frac{726415849090107053}{35852671903975243944} a^{6} - \frac{19462298120376959}{103024919264296678} a^{4} + \frac{130695419292301}{322962129355162} a^{2} + \frac{1808697128483}{4049681872792}$, $\frac{1}{286319437825146298136784} a^{15} - \frac{1}{26029039802286027103344} a^{14} + \frac{2545836239299}{31813270869460699792976} a^{13} + \frac{6404322993197}{8676346600762009034448} a^{12} + \frac{14412575703899303}{2892115533587336344816} a^{11} + \frac{2072710759691321}{788758781887455366768} a^{10} - \frac{195514809492182057}{8676346600762009034448} a^{9} - \frac{52604356842848923}{788758781887455366768} a^{8} + \frac{13701700232567962283}{788758781887455366768} a^{7} + \frac{2215433421904508065}{71705343807950487888} a^{6} - \frac{2212793253790367153}{9066192895258107664} a^{5} + \frac{394833617237038423}{824199354114373424} a^{4} + \frac{42025447894087}{112334653688752} a^{3} + \frac{138967153641867}{2583697034841296} a^{2} + \frac{31829164924615}{89093001201424} a - \frac{136002441895}{8099363745584}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 51844297037900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.13824.1, 4.4.76032.1, 4.4.50688.2, 8.8.369975361536.2

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$