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Properties

Label 16.0.27404580206...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{10}\cdot 761^{5}$
Root discriminant $122.99$
Ramified primes $2, 5, 761$
Class number $1142048$ (GRH)
Class group $[2, 2, 2, 142756]$ (GRH)
Galois group 16T1360

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11017777025, 0, 7673353250, 0, 1901560165, 0, 237751620, 0, 16860739, 0, 700728, 0, 16665, 0, 206, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 206*x^14 + 16665*x^12 + 700728*x^10 + 16860739*x^8 + 237751620*x^6 + 1901560165*x^4 + 7673353250*x^2 + 11017777025)
 
gp: K = bnfinit(x^16 + 206*x^14 + 16665*x^12 + 700728*x^10 + 16860739*x^8 + 237751620*x^6 + 1901560165*x^4 + 7673353250*x^2 + 11017777025, 1)
 

Normalized defining polynomial

\( x^{16} + 206 x^{14} + 16665 x^{12} + 700728 x^{10} + 16860739 x^{8} + 237751620 x^{6} + 1901560165 x^{4} + 7673353250 x^{2} + 11017777025 \)

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:  \(2740458020629184239370240000000000=2^{40}\cdot 5^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.99$
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:  $2, 5, 761$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3805} a^{12} + \frac{206}{3805} a^{10} + \frac{289}{761} a^{8} + \frac{608}{3805} a^{6} + \frac{784}{3805} a^{4}$, $\frac{1}{3805} a^{13} + \frac{206}{3805} a^{11} + \frac{289}{761} a^{9} + \frac{608}{3805} a^{7} + \frac{784}{3805} a^{5}$, $\frac{1}{3759540855075874040807115} a^{14} - \frac{118203956593565164429}{1253180285025291346935705} a^{12} - \frac{24174079125885094689352}{54486099348925710736335} a^{10} + \frac{62671363912561582885347}{417726761675097115645235} a^{8} - \frac{151699655891243855412442}{751908171015174808161423} a^{6} + \frac{42399210531773947}{963202171127069835} a^{4} + \frac{359162747438108125544}{988052787142148236743} a^{2} + \frac{52324111143248885}{1298361086914780863}$, $\frac{1}{3759540855075874040807115} a^{15} - \frac{118203956593565164429}{1253180285025291346935705} a^{13} - \frac{24174079125885094689352}{54486099348925710736335} a^{11} + \frac{62671363912561582885347}{417726761675097115645235} a^{9} - \frac{151699655891243855412442}{751908171015174808161423} a^{7} + \frac{42399210531773947}{963202171127069835} a^{5} + \frac{359162747438108125544}{988052787142148236743} a^{3} + \frac{52324111143248885}{1298361086914780863} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{142756}$, which has order $1142048$ (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:  \( 9562.41394678 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1360:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 65 conjugacy class representatives for t16n1360 are not computed
Character table for t16n1360 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
761Data not computed