↑

Properties

Label 16.8.85830270230...6489.4
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 $C_2^2.D_4$ (as 16T33)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![208, -728, 192, 5028, -331, -9723, -1418, 5648, -149, -1008, 813, 33, -141, 49, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + x^14 + 49*x^13 - 141*x^12 + 33*x^11 + 813*x^10 - 1008*x^9 - 149*x^8 + 5648*x^7 - 1418*x^6 - 9723*x^5 - 331*x^4 + 5028*x^3 + 192*x^2 - 728*x + 208)
 
gp: K = bnfinit(x^16 - 6*x^15 + x^14 + 49*x^13 - 141*x^12 + 33*x^11 + 813*x^10 - 1008*x^9 - 149*x^8 + 5648*x^7 - 1418*x^6 - 9723*x^5 - 331*x^4 + 5028*x^3 + 192*x^2 - 728*x + 208, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + x^{14} + 49 x^{13} - 141 x^{12} + 33 x^{11} + 813 x^{10} - 1008 x^{9} - 149 x^{8} + 5648 x^{7} - 1418 x^{6} - 9723 x^{5} - 331 x^{4} + 5028 x^{3} + 192 x^{2} - 728 x + 208 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{1}{18} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{5}{18} a + \frac{2}{9}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} + \frac{1}{18} a^{8} - \frac{1}{18} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{5}{18} a^{2} + \frac{2}{9} a$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{10} + \frac{2}{27} a^{9} + \frac{1}{54} a^{8} - \frac{1}{27} a^{7} - \frac{2}{27} a^{6} - \frac{1}{54} a^{4} - \frac{5}{54} a^{3} + \frac{7}{54} a^{2} - \frac{11}{54} a - \frac{13}{27}$, $\frac{1}{594} a^{13} - \frac{1}{594} a^{12} - \frac{13}{594} a^{11} - \frac{8}{297} a^{10} + \frac{8}{99} a^{9} - \frac{1}{22} a^{8} + \frac{49}{594} a^{7} + \frac{49}{594} a^{6} + \frac{37}{297} a^{5} + \frac{148}{297} a^{4} - \frac{1}{18} a^{3} - \frac{61}{198} a^{2} - \frac{31}{198} a - \frac{74}{297}$, $\frac{1}{566676} a^{14} + \frac{1}{1431} a^{13} + \frac{395}{188892} a^{12} - \frac{3659}{566676} a^{11} + \frac{5155}{188892} a^{10} - \frac{17821}{566676} a^{9} - \frac{625}{17172} a^{8} - \frac{12535}{283338} a^{7} - \frac{85655}{566676} a^{6} - \frac{47203}{283338} a^{5} + \frac{15469}{31482} a^{4} + \frac{144665}{566676} a^{3} - \frac{221233}{566676} a^{2} - \frac{52874}{141669} a - \frac{17483}{141669}$, $\frac{1}{10651177494936} a^{15} + \frac{3221209}{5325588747468} a^{14} + \frac{731310995}{3550392498312} a^{13} - \frac{90667253855}{10651177494936} a^{12} + \frac{11764007975}{10651177494936} a^{11} - \frac{179094146503}{10651177494936} a^{10} - \frac{201073663511}{10651177494936} a^{9} - \frac{14895252721}{91820495646} a^{8} - \frac{103829626039}{3550392498312} a^{7} - \frac{15861976303}{147933020763} a^{6} - \frac{145088705723}{5325588747468} a^{5} + \frac{223739796473}{10651177494936} a^{4} - \frac{185940211001}{3550392498312} a^{3} - \frac{346274070881}{1331397186867} a^{2} + \frac{13699336198}{443799062289} a + \frac{463037314117}{1331397186867}$

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:  \( 9241015.28903 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), 4.4.8957.1 x2, 4.4.36517.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.2929680361933.4 x2, 8.4.225360027841.1 x2, 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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ 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$13.2.1.1$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}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$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}$