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Properties

Label 16.4.39539538769...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 11^{3}\cdot 106759^{3}$
Root discriminant $45.95$
Ramified primes $5, 11, 106759$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1714129, 3560555, -2972302, 2014235, -1310594, 665450, -292336, 98025, -19871, 4845, -138, -680, 94, -55, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 21*x^14 - 55*x^13 + 94*x^12 - 680*x^11 - 138*x^10 + 4845*x^9 - 19871*x^8 + 98025*x^7 - 292336*x^6 + 665450*x^5 - 1310594*x^4 + 2014235*x^3 - 2972302*x^2 + 3560555*x - 1714129)
 
gp: K = bnfinit(x^16 + 21*x^14 - 55*x^13 + 94*x^12 - 680*x^11 - 138*x^10 + 4845*x^9 - 19871*x^8 + 98025*x^7 - 292336*x^6 + 665450*x^5 - 1310594*x^4 + 2014235*x^3 - 2972302*x^2 + 3560555*x - 1714129, 1)
 

Normalized defining polynomial

\( x^{16} + 21 x^{14} - 55 x^{13} + 94 x^{12} - 680 x^{11} - 138 x^{10} + 4845 x^{9} - 19871 x^{8} + 98025 x^{7} - 292336 x^{6} + 665450 x^{5} - 1310594 x^{4} + 2014235 x^{3} - 2972302 x^{2} + 3560555 x - 1714129 \)

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:  \(395395387694734020751953125=5^{12}\cdot 11^{3}\cdot 106759^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.95$
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:  $5, 11, 106759$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12605790531761211094908373372340179040425104671} a^{15} - \frac{3207034577492418160242952856933308655763526206}{12605790531761211094908373372340179040425104671} a^{14} + \frac{1790929496444088725561432508045028688607927169}{12605790531761211094908373372340179040425104671} a^{13} + \frac{6224500129944850640578360543158850105525477982}{12605790531761211094908373372340179040425104671} a^{12} - \frac{4173030948723052879720369852603541910395821311}{12605790531761211094908373372340179040425104671} a^{11} + \frac{3976233465570026165753944740278230281014167622}{12605790531761211094908373372340179040425104671} a^{10} - \frac{1876427550932797533386259128209822402544484889}{12605790531761211094908373372340179040425104671} a^{9} + \frac{75198017233305894834530202703209444667882922}{12605790531761211094908373372340179040425104671} a^{8} + \frac{5550208675075559991014812223298048350635037331}{12605790531761211094908373372340179040425104671} a^{7} + \frac{5804211471329689648708261339669571469264194299}{12605790531761211094908373372340179040425104671} a^{6} - \frac{850727678703140340250988655441010909478042485}{12605790531761211094908373372340179040425104671} a^{5} - \frac{4603160230286511260343417993931586278540016475}{12605790531761211094908373372340179040425104671} a^{4} - \frac{1120979568178751097064670863312265606464264163}{12605790531761211094908373372340179040425104671} a^{3} + \frac{5740943573693492946963517109187659045994229349}{12605790531761211094908373372340179040425104671} a^{2} + \frac{5151276160159934245173106292309097088086103781}{12605790531761211094908373372340179040425104671} a - \frac{2062170884668789160246534792603134379904574762}{12605790531761211094908373372340179040425104671}$

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:  \( 3273733.6185 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.733968125.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R $16$ $16$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
106759Data not computed