↑

Properties

Label 16.0.11662101038...3641.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{6}\cdot 37^{12}$
Root discriminant $36.87$
Ramified primes $11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1913, 2527, 5422, 7300, 8911, 5823, 3221, 1425, 1144, 545, 260, 65, 42, -7, 14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 14*x^14 - 7*x^13 + 42*x^12 + 65*x^11 + 260*x^10 + 545*x^9 + 1144*x^8 + 1425*x^7 + 3221*x^6 + 5823*x^5 + 8911*x^4 + 7300*x^3 + 5422*x^2 + 2527*x + 1913)
 
gp: K = bnfinit(x^16 - 3*x^15 + 14*x^14 - 7*x^13 + 42*x^12 + 65*x^11 + 260*x^10 + 545*x^9 + 1144*x^8 + 1425*x^7 + 3221*x^6 + 5823*x^5 + 8911*x^4 + 7300*x^3 + 5422*x^2 + 2527*x + 1913, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 14 x^{14} - 7 x^{13} + 42 x^{12} + 65 x^{11} + 260 x^{10} + 545 x^{9} + 1144 x^{8} + 1425 x^{7} + 3221 x^{6} + 5823 x^{5} + 8911 x^{4} + 7300 x^{3} + 5422 x^{2} + 2527 x + 1913 \)

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:  \(11662101038417978742443641=11^{6}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.87$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{1048002590366201489878327} a^{15} + \frac{48803444101232177362294}{1048002590366201489878327} a^{14} - \frac{59241180017154787183354}{1048002590366201489878327} a^{13} + \frac{71638774726982217805625}{1048002590366201489878327} a^{12} - \frac{65930413335702786290898}{1048002590366201489878327} a^{11} + \frac{72955154260512077253972}{1048002590366201489878327} a^{10} + \frac{478412173051472791885981}{1048002590366201489878327} a^{9} + \frac{29102916638801374168809}{149714655766600212839761} a^{8} + \frac{340069373551072158686563}{1048002590366201489878327} a^{7} + \frac{176409234110647654299839}{1048002590366201489878327} a^{6} + \frac{171378981035033786507610}{1048002590366201489878327} a^{5} + \frac{64002196921035124985901}{1048002590366201489878327} a^{4} - \frac{331832812450574372320298}{1048002590366201489878327} a^{3} + \frac{124496648346895957984576}{1048002590366201489878327} a^{2} - \frac{9419740691025674427037}{1048002590366201489878327} a + \frac{216077342128015398237}{547831986600209874479}$

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:  $7$
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:  \( 840629.139795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.2.15059.1, 4.0.50653.1, 4.2.557183.1, 8.2.3414981850379.1, 8.2.2494508291.1, 8.0.310452895489.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$