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Properties

Label 16.8.13735096585...9809.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{8}\cdot 17^{14}$
Root discriminant $43.01$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, 2522, -6815, 2116, 3369, 2780, -2096, -3096, 334, 864, 486, -44, -124, 14, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + x^14 + 14*x^13 - 124*x^12 - 44*x^11 + 486*x^10 + 864*x^9 + 334*x^8 - 3096*x^7 - 2096*x^6 + 2780*x^5 + 3369*x^4 + 2116*x^3 - 6815*x^2 + 2522*x - 52)
 
gp: K = bnfinit(x^16 - 4*x^15 + x^14 + 14*x^13 - 124*x^12 - 44*x^11 + 486*x^10 + 864*x^9 + 334*x^8 - 3096*x^7 - 2096*x^6 + 2780*x^5 + 3369*x^4 + 2116*x^3 - 6815*x^2 + 2522*x - 52, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + x^{14} + 14 x^{13} - 124 x^{12} - 44 x^{11} + 486 x^{10} + 864 x^{9} + 334 x^{8} - 3096 x^{7} - 2096 x^{6} + 2780 x^{5} + 3369 x^{4} + 2116 x^{3} - 6815 x^{2} + 2522 x - 52 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(137350965859713069141239809=13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.01$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{61408} a^{14} + \frac{141}{30704} a^{13} + \frac{881}{15352} a^{12} - \frac{487}{15352} a^{11} - \frac{1}{3838} a^{10} - \frac{919}{7676} a^{9} + \frac{3471}{30704} a^{8} - \frac{3475}{15352} a^{7} - \frac{377}{1919} a^{6} + \frac{583}{15352} a^{5} + \frac{7}{101} a^{4} - \frac{3039}{7676} a^{3} + \frac{33}{61408} a^{2} + \frac{881}{30704} a - \frac{1581}{15352}$, $\frac{1}{743497373334501568} a^{15} + \frac{435341924855}{743497373334501568} a^{14} - \frac{101421642566911}{371748686667250784} a^{13} - \frac{4483391598879843}{185874343333625392} a^{12} + \frac{462029803380479}{11617146458351587} a^{11} + \frac{22237860017418153}{185874343333625392} a^{10} + \frac{33291186425105373}{371748686667250784} a^{9} - \frac{15030638426102257}{371748686667250784} a^{8} - \frac{10868548130061173}{46468585833406348} a^{7} + \frac{32612728370307}{611428760965873} a^{6} + \frac{2353639343153923}{23234292916703174} a^{5} + \frac{25375940868871195}{185874343333625392} a^{4} + \frac{35742117614018533}{743497373334501568} a^{3} - \frac{119114482673735893}{743497373334501568} a^{2} + \frac{46049708941885599}{371748686667250784} a - \frac{8367979353604877}{185874343333625392}$

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:  $11$
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:  \( 56939786.1099 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{221}) \), 4.4.4913.1, 4.4.830297.1, \(\Q(\sqrt{13}, \sqrt{17})\), 8.4.69347235737.1 x2, 8.4.11719682839553.1 x2, 8.8.689393108209.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$