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Properties

Label 16.4.75610373616...437.13
Degree $16$
Signature $[4, 6]$
Discriminant $13^{8}\cdot 53^{11}$
Root discriminant $55.26$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 507, 3874, 13468, -26571, 22642, -15512, 8838, -5446, 2830, -532, -174, 110, -18, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 18*x^13 + 110*x^12 - 174*x^11 - 532*x^10 + 2830*x^9 - 5446*x^8 + 8838*x^7 - 15512*x^6 + 22642*x^5 - 26571*x^4 + 13468*x^3 + 3874*x^2 + 507*x + 169)
 
gp: K = bnfinit(x^16 - 3*x^15 - 18*x^13 + 110*x^12 - 174*x^11 - 532*x^10 + 2830*x^9 - 5446*x^8 + 8838*x^7 - 15512*x^6 + 22642*x^5 - 26571*x^4 + 13468*x^3 + 3874*x^2 + 507*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 18 x^{13} + 110 x^{12} - 174 x^{11} - 532 x^{10} + 2830 x^{9} - 5446 x^{8} + 8838 x^{7} - 15512 x^{6} + 22642 x^{5} - 26571 x^{4} + 13468 x^{3} + 3874 x^{2} + 507 x + 169 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7561037361641682928770951437=13^{8}\cdot 53^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.26$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{1}{13} a^{10} - \frac{2}{13} a^{9} + \frac{1}{13} a^{8} - \frac{4}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{13} a^{5} - \frac{6}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{13} + \frac{1}{13} a^{11} - \frac{2}{13} a^{10} + \frac{1}{13} a^{9} - \frac{4}{13} a^{8} + \frac{3}{13} a^{7} + \frac{1}{13} a^{6} - \frac{6}{13} a^{5} - \frac{6}{13} a^{4} - \frac{1}{13} a^{3}$, $\frac{1}{39} a^{14} - \frac{1}{39} a^{13} - \frac{1}{39} a^{12} - \frac{1}{13} a^{11} + \frac{1}{39} a^{10} - \frac{1}{39} a^{9} - \frac{8}{39} a^{8} - \frac{7}{39} a^{7} + \frac{1}{3} a^{6} - \frac{2}{39} a^{5} + \frac{4}{39} a^{4} + \frac{1}{3} a^{3} + \frac{5}{13} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{55081247622671903375469102519} a^{15} - \frac{163049729193657317781988835}{55081247622671903375469102519} a^{14} - \frac{126789832663823337637246945}{18360415874223967791823034173} a^{13} + \frac{52947362863312834982427749}{5007386147515627579588100229} a^{12} - \frac{1427722176591244094719021867}{5007386147515627579588100229} a^{11} + \frac{24122678688202090574488188907}{55081247622671903375469102519} a^{10} + \frac{417631304778177692166173317}{5007386147515627579588100229} a^{9} + \frac{17171540235204449133286062412}{55081247622671903375469102519} a^{8} + \frac{10563282037937074365774053006}{55081247622671903375469102519} a^{7} + \frac{128948959326503629239941066}{1412339682632612907063310321} a^{6} + \frac{482726888154752120715649870}{1412339682632612907063310321} a^{5} + \frac{7438078443880331143584737245}{18360415874223967791823034173} a^{4} + \frac{10772134253391843893505979412}{55081247622671903375469102519} a^{3} - \frac{2424126161210563056209893204}{55081247622671903375469102519} a^{2} - \frac{355285052504218715157897137}{1412339682632612907063310321} a + \frac{1100934330761725529191388188}{4237019047897838721189930963}$

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:  $9$
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:  \( 46843154.1664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 8.4.11944081475573.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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}$
13.2.1.1$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}$
53Data not computed