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Properties

Label 16.0.14505315669...641.25
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $64$ (GRH)
Class group $[2, 2, 16]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![374959, -649142, 403234, 112359, -226678, 75155, 90672, -78107, 31820, -1340, -639, 125, 272, -39, 37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 37*x^14 - 39*x^13 + 272*x^12 + 125*x^11 - 639*x^10 - 1340*x^9 + 31820*x^8 - 78107*x^7 + 90672*x^6 + 75155*x^5 - 226678*x^4 + 112359*x^3 + 403234*x^2 - 649142*x + 374959)
 
gp: K = bnfinit(x^16 - 2*x^15 + 37*x^14 - 39*x^13 + 272*x^12 + 125*x^11 - 639*x^10 - 1340*x^9 + 31820*x^8 - 78107*x^7 + 90672*x^6 + 75155*x^5 - 226678*x^4 + 112359*x^3 + 403234*x^2 - 649142*x + 374959, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 37 x^{14} - 39 x^{13} + 272 x^{12} + 125 x^{11} - 639 x^{10} - 1340 x^{9} + 31820 x^{8} - 78107 x^{7} + 90672 x^{6} + 75155 x^{5} - 226678 x^{4} + 112359 x^{3} + 403234 x^{2} - 649142 x + 374959 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{174860406060230635166006591955017967604359} a^{15} + \frac{8395667743482850330196051226313705767218}{174860406060230635166006591955017967604359} a^{14} + \frac{28976688513355646529410394964209036339040}{174860406060230635166006591955017967604359} a^{13} - \frac{27961938788244247274267249588929924168579}{174860406060230635166006591955017967604359} a^{12} - \frac{26099337475273170737803335990596226909224}{174860406060230635166006591955017967604359} a^{11} + \frac{21227671729755004117121007410850308372561}{174860406060230635166006591955017967604359} a^{10} - \frac{6963121200381406528637816889307212590066}{174860406060230635166006591955017967604359} a^{9} + \frac{1905201704319514804499209895659903210259}{174860406060230635166006591955017967604359} a^{8} + \frac{7340966361451543937305907582910989536397}{174860406060230635166006591955017967604359} a^{7} + \frac{76269135648092671538836380509675433169429}{174860406060230635166006591955017967604359} a^{6} - \frac{19412115056134824212960610349408799563075}{174860406060230635166006591955017967604359} a^{5} + \frac{83338667768231621679650457623335728251980}{174860406060230635166006591955017967604359} a^{4} + \frac{65146710815892866083429137214893381875132}{174860406060230635166006591955017967604359} a^{3} - \frac{78517662253628204954987622583414835660345}{174860406060230635166006591955017967604359} a^{2} + \frac{56519907516186957970948405139754517862399}{174860406060230635166006591955017967604359} a + \frac{2673384011658286437911018626489767794986}{174860406060230635166006591955017967604359}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{16}$, which has order $64$ (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:  $7$
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:  \( 304116.798972 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.6171373.2, 4.4.8957.1, 4.0.116441.1, 8.4.2929680361933.1, 8.4.1042962037.1, 8.0.38085844705129.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
53Data not computed