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Properties

Label 16.4.75610373616...437.19
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![10331, -10083, -4011, 2957, 6552, 18493, 8161, -3270, -2932, 541, 161, -256, -42, 45, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 + 45*x^13 - 42*x^12 - 256*x^11 + 161*x^10 + 541*x^9 - 2932*x^8 - 3270*x^7 + 8161*x^6 + 18493*x^5 + 6552*x^4 + 2957*x^3 - 4011*x^2 - 10083*x + 10331)
 
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 + 45*x^13 - 42*x^12 - 256*x^11 + 161*x^10 + 541*x^9 - 2932*x^8 - 3270*x^7 + 8161*x^6 + 18493*x^5 + 6552*x^4 + 2957*x^3 - 4011*x^2 - 10083*x + 10331, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 4 x^{14} + 45 x^{13} - 42 x^{12} - 256 x^{11} + 161 x^{10} + 541 x^{9} - 2932 x^{8} - 3270 x^{7} + 8161 x^{6} + 18493 x^{5} + 6552 x^{4} + 2957 x^{3} - 4011 x^{2} - 10083 x + 10331 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{9} a^{10} + \frac{11}{27} a^{9} - \frac{4}{9} a^{8} - \frac{5}{27} a^{7} + \frac{5}{27} a^{6} + \frac{1}{27} a^{5} - \frac{13}{27} a^{4} - \frac{1}{27} a^{3} - \frac{10}{27} a^{2} + \frac{2}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{9} a^{11} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{13}{27} a^{4} + \frac{7}{27} a^{3} - \frac{8}{27} a^{2} - \frac{11}{27}$, $\frac{1}{232721028914313700396980594117} a^{15} - \frac{3412445462192134763567413921}{232721028914313700396980594117} a^{14} - \frac{1001676956370713977452567271}{232721028914313700396980594117} a^{13} - \frac{1286815216390722023610839294}{33245861273473385770997227731} a^{12} - \frac{5102644819268986584114858625}{33245861273473385770997227731} a^{11} + \frac{361376534296552514278811159}{2873099122398934572802229557} a^{10} - \frac{70988977120707821238491251657}{232721028914313700396980594117} a^{9} + \frac{13013932278099980379481971188}{33245861273473385770997227731} a^{8} + \frac{153442398693283986971514682}{8619297367196803718406688671} a^{7} - \frac{2158891591663182063716651065}{77573676304771233465660198039} a^{6} + \frac{71574858473780143720241388757}{232721028914313700396980594117} a^{5} + \frac{3347469848115178713493639973}{25857892101590411155220066013} a^{4} + \frac{156370887984846105406162394}{2873099122398934572802229557} a^{3} + \frac{3338086614311503815804735617}{33245861273473385770997227731} a^{2} + \frac{2911185735075430553982798034}{33245861273473385770997227731} a - \frac{61132061328284472241746893845}{232721028914313700396980594117}$

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:  \( 50253405.9096 \) (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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ 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$13.4.2.1$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}$
13.4.2.1$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}$
53Data not computed