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Properties

Label 16.0.98855585443...5424.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 11^{8}$
Root discriminant $48.66$
Ramified primes $2, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![463, -4536, 20220, -54592, 100122, -133032, 133904, -105688, 67067, -34408, 14784, -5080, 1514, -336, 68, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1514*x^12 - 5080*x^11 + 14784*x^10 - 34408*x^9 + 67067*x^8 - 105688*x^7 + 133904*x^6 - 133032*x^5 + 100122*x^4 - 54592*x^3 + 20220*x^2 - 4536*x + 463)
 
gp: K = bnfinit(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1514*x^12 - 5080*x^11 + 14784*x^10 - 34408*x^9 + 67067*x^8 - 105688*x^7 + 133904*x^6 - 133032*x^5 + 100122*x^4 - 54592*x^3 + 20220*x^2 - 4536*x + 463, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 68 x^{14} - 336 x^{13} + 1514 x^{12} - 5080 x^{11} + 14784 x^{10} - 34408 x^{9} + 67067 x^{8} - 105688 x^{7} + 133904 x^{6} - 133032 x^{5} + 100122 x^{4} - 54592 x^{3} + 20220 x^{2} - 4536 x + 463 \)

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:  \(988555854433440250854375424=2^{62}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.66$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(352=2^{5}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{352}(1,·)$, $\chi_{352}(67,·)$, $\chi_{352}(65,·)$, $\chi_{352}(265,·)$, $\chi_{352}(331,·)$, $\chi_{352}(131,·)$, $\chi_{352}(89,·)$, $\chi_{352}(153,·)$, $\chi_{352}(155,·)$, $\chi_{352}(219,·)$, $\chi_{352}(243,·)$, $\chi_{352}(241,·)$, $\chi_{352}(43,·)$, $\chi_{352}(177,·)$, $\chi_{352}(307,·)$, $\chi_{352}(329,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} + \frac{3}{17} a^{10} + \frac{5}{17} a^{9} + \frac{6}{17} a^{8} - \frac{1}{17} a^{7} + \frac{5}{17} a^{6} - \frac{5}{17} a^{4} - \frac{6}{17} a^{3} - \frac{6}{17} a - \frac{1}{17}$, $\frac{1}{17} a^{12} - \frac{4}{17} a^{10} + \frac{8}{17} a^{9} - \frac{2}{17} a^{8} + \frac{8}{17} a^{7} + \frac{2}{17} a^{6} - \frac{5}{17} a^{5} - \frac{8}{17} a^{4} + \frac{1}{17} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17}$, $\frac{1}{17} a^{13} + \frac{3}{17} a^{10} + \frac{1}{17} a^{9} - \frac{2}{17} a^{8} - \frac{2}{17} a^{7} - \frac{2}{17} a^{6} - \frac{8}{17} a^{5} - \frac{2}{17} a^{4} + \frac{4}{17} a^{3} - \frac{4}{17} a - \frac{4}{17}$, $\frac{1}{211307807} a^{14} - \frac{7}{211307807} a^{13} - \frac{1759082}{211307807} a^{12} - \frac{1875288}{211307807} a^{11} - \frac{103060821}{211307807} a^{10} - \frac{2432449}{12429871} a^{9} - \frac{86701018}{211307807} a^{8} + \frac{52231406}{211307807} a^{7} + \frac{39657776}{211307807} a^{6} - \frac{8599871}{211307807} a^{5} - \frac{74564626}{211307807} a^{4} - \frac{67821205}{211307807} a^{3} - \frac{8946042}{211307807} a^{2} + \frac{66622861}{211307807} a - \frac{75587497}{211307807}$, $\frac{1}{211307807} a^{15} - \frac{1759131}{211307807} a^{13} - \frac{1758991}{211307807} a^{12} - \frac{4318998}{211307807} a^{11} - \frac{54274733}{211307807} a^{10} + \frac{71312907}{211307807} a^{9} + \frac{91677572}{211307807} a^{8} - \frac{1751051}{12429871} a^{7} + \frac{7977270}{211307807} a^{6} + \frac{14394729}{211307807} a^{5} + \frac{19290092}{211307807} a^{4} - \frac{885965}{2178431} a^{3} - \frac{70578659}{211307807} a^{2} - \frac{69132697}{211307807} a + \frac{30231716}{211307807}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 1365751.160205918 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\zeta_{16})^+\), 4.0.247808.2, 8.0.61408804864.2, 8.8.31441308090368.1, 8.0.2147483648.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
11Data not computed