↑

Properties

Label 16.8.85830270230...6489.3
Degree $16$
Signature $[8, 4]$
Discriminant $13^{10}\cdot 53^{8}$
Root discriminant $36.17$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 143, 431, -1223, 2574, 493, -7994, -3145, 1264, 1444, -5, -265, 90, 35, -19, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121)
 
gp: K = bnfinit(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 19 x^{14} + 35 x^{13} + 90 x^{12} - 265 x^{11} - 5 x^{10} + 1444 x^{9} + 1264 x^{8} - 3145 x^{7} - 7994 x^{6} + 493 x^{5} + 2574 x^{4} - 1223 x^{3} + 431 x^{2} + 143 x + 121 \)

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:  \(8583027023095873875496489=13^{10}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.17$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{39} a^{12} + \frac{4}{39} a^{11} - \frac{2}{39} a^{10} + \frac{2}{39} a^{9} + \frac{5}{13} a^{7} - \frac{1}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{39} a^{4} - \frac{10}{39} a^{3} + \frac{14}{39} a^{2} - \frac{11}{39} a - \frac{3}{13}$, $\frac{1}{39} a^{13} - \frac{5}{39} a^{11} - \frac{1}{13} a^{10} + \frac{5}{39} a^{9} + \frac{2}{39} a^{8} + \frac{5}{13} a^{7} + \frac{6}{13} a^{6} + \frac{11}{39} a^{5} + \frac{2}{13} a^{4} + \frac{2}{39} a^{3} - \frac{5}{13} a^{2} - \frac{17}{39} a + \frac{10}{39}$, $\frac{1}{1521} a^{14} + \frac{14}{1521} a^{13} + \frac{11}{1521} a^{12} + \frac{10}{507} a^{11} - \frac{43}{1521} a^{10} - \frac{14}{117} a^{9} - \frac{152}{1521} a^{8} + \frac{3}{13} a^{7} + \frac{371}{1521} a^{6} + \frac{100}{1521} a^{5} + \frac{724}{1521} a^{4} - \frac{244}{507} a^{3} - \frac{536}{1521} a^{2} - \frac{625}{1521} a - \frac{394}{1521}$, $\frac{1}{333746979150890822517} a^{15} + \frac{13577286300560813}{47678139878698688931} a^{14} + \frac{222415876524395843}{333746979150890822517} a^{13} - \frac{357089620901602240}{111248993050296940839} a^{12} + \frac{936414021730116572}{47678139878698688931} a^{11} - \frac{24338590299287020868}{333746979150890822517} a^{10} - \frac{25592336994848893538}{333746979150890822517} a^{9} + \frac{1677062550612429391}{37082997683432313613} a^{8} + \frac{30221605072455978422}{333746979150890822517} a^{7} + \frac{58556507905405462789}{333746979150890822517} a^{6} + \frac{24123494885992042192}{333746979150890822517} a^{5} - \frac{40413419650969482554}{111248993050296940839} a^{4} + \frac{12614939913943183952}{30340634468262802047} a^{3} - \frac{13347954265767810328}{333746979150890822517} a^{2} - \frac{43154301020366019493}{333746979150890822517} a + \frac{1621616366247666494}{3371181607584755783}$

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:  $11$
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:  \( 6129504.74347 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{689}) \), 4.4.36517.1 x2, 4.4.8957.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.17335386757.1, 8.4.2929680361933.2, 8.8.225360027841.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$