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Properties

Label 16.0.35847274805...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{8}$
Root discriminant $25.68$
Ramified primes $5, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 58, 485, 608, 952, -195, 487, -185, 658, -246, 265, -116, 66, -11, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 13*x^14 - 11*x^13 + 66*x^12 - 116*x^11 + 265*x^10 - 246*x^9 + 658*x^8 - 185*x^7 + 487*x^6 - 195*x^5 + 952*x^4 + 608*x^3 + 485*x^2 + 58*x + 11)
 
gp: K = bnfinit(x^16 + 13*x^14 - 11*x^13 + 66*x^12 - 116*x^11 + 265*x^10 - 246*x^9 + 658*x^8 - 185*x^7 + 487*x^6 - 195*x^5 + 952*x^4 + 608*x^3 + 485*x^2 + 58*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} + 13 x^{14} - 11 x^{13} + 66 x^{12} - 116 x^{11} + 265 x^{10} - 246 x^{9} + 658 x^{8} - 185 x^{7} + 487 x^{6} - 195 x^{5} + 952 x^{4} + 608 x^{3} + 485 x^{2} + 58 x + 11 \)

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:  \(35847274805742431640625=5^{12}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{15} a^{14} - \frac{2}{15} a^{13} + \frac{1}{15} a^{12} + \frac{4}{15} a^{11} + \frac{7}{15} a^{10} + \frac{1}{15} a^{9} - \frac{4}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{6} + \frac{4}{15} a^{5} - \frac{2}{5} a^{4} - \frac{2}{15} a^{3} + \frac{7}{15} a^{2} + \frac{1}{15} a + \frac{2}{5}$, $\frac{1}{3500411223440440995015} a^{15} + \frac{9895436533336419688}{1166803741146813665005} a^{14} - \frac{408085601100576412186}{3500411223440440995015} a^{13} + \frac{11585875602090572312}{233360748229362733001} a^{12} + \frac{51063856977833474096}{3500411223440440995015} a^{11} + \frac{383616980310332203916}{1166803741146813665005} a^{10} - \frac{139336106094172474491}{1166803741146813665005} a^{9} + \frac{1375017244567359963347}{3500411223440440995015} a^{8} + \frac{1153028121407369823826}{3500411223440440995015} a^{7} - \frac{1621108009709013597056}{3500411223440440995015} a^{6} - \frac{528790136070114240149}{1166803741146813665005} a^{5} + \frac{83318159851476465422}{3500411223440440995015} a^{4} + \frac{349664845636004508886}{700082244688088199003} a^{3} + \frac{114605000029537721421}{1166803741146813665005} a^{2} + \frac{659410072576738869202}{3500411223440440995015} a + \frac{17926787662672782957}{106073067376983060455}$

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:  \( -1 \) (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:  \( 30730.9014136 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.0.37866753125.1, 8.2.3209046875.2, 8.2.641809375.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$