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Properties

Label 16.16.3761893960...8784.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{62}\cdot 13^{8}$
Root discriminant $52.90$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (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![127, -424, -2564, 13904, -16214, -14600, 36768, -7240, -19381, 8312, 3600, -2184, -182, 224, -12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 12*x^14 + 224*x^13 - 182*x^12 - 2184*x^11 + 3600*x^10 + 8312*x^9 - 19381*x^8 - 7240*x^7 + 36768*x^6 - 14600*x^5 - 16214*x^4 + 13904*x^3 - 2564*x^2 - 424*x + 127)
 
gp: K = bnfinit(x^16 - 8*x^15 - 12*x^14 + 224*x^13 - 182*x^12 - 2184*x^11 + 3600*x^10 + 8312*x^9 - 19381*x^8 - 7240*x^7 + 36768*x^6 - 14600*x^5 - 16214*x^4 + 13904*x^3 - 2564*x^2 - 424*x + 127, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 12 x^{14} + 224 x^{13} - 182 x^{12} - 2184 x^{11} + 3600 x^{10} + 8312 x^{9} - 19381 x^{8} - 7240 x^{7} + 36768 x^{6} - 14600 x^{5} - 16214 x^{4} + 13904 x^{3} - 2564 x^{2} - 424 x + 127 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3761893960837392421076598784=2^{62}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.90$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(416=2^{5}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(389,·)$, $\chi_{416}(129,·)$, $\chi_{416}(77,·)$, $\chi_{416}(337,·)$, $\chi_{416}(25,·)$, $\chi_{416}(157,·)$, $\chi_{416}(261,·)$, $\chi_{416}(209,·)$, $\chi_{416}(233,·)$, $\chi_{416}(365,·)$, $\chi_{416}(285,·)$, $\chi_{416}(53,·)$, $\chi_{416}(105,·)$, $\chi_{416}(313,·)$, $\chi_{416}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5490568271} a^{14} - \frac{7}{5490568271} a^{13} - \frac{708359885}{5490568271} a^{12} - \frac{1240408870}{5490568271} a^{11} + \frac{676200516}{5490568271} a^{10} + \frac{1583748912}{5490568271} a^{9} - \frac{207729399}{5490568271} a^{8} - \frac{353800076}{5490568271} a^{7} - \frac{380467942}{5490568271} a^{6} + \frac{164616881}{5490568271} a^{5} - \frac{1685707149}{5490568271} a^{4} - \frac{1814892112}{5490568271} a^{3} - \frac{285968309}{5490568271} a^{2} - \frac{1237800832}{5490568271} a + \frac{562804569}{5490568271}$, $\frac{1}{2542133109473} a^{15} + \frac{224}{2542133109473} a^{14} - \frac{544274620331}{2542133109473} a^{13} - \frac{439399955855}{2542133109473} a^{12} + \frac{1267972572239}{2542133109473} a^{11} - \frac{918365313008}{2542133109473} a^{10} - \frac{611682882965}{2542133109473} a^{9} - \frac{1025660443483}{2542133109473} a^{8} + \frac{1065420483141}{2542133109473} a^{7} - \frac{1218780541547}{2542133109473} a^{6} - \frac{369961259692}{2542133109473} a^{5} - \frac{748100181146}{2542133109473} a^{4} + \frac{552304537786}{2542133109473} a^{3} + \frac{1014345469176}{2542133109473} a^{2} - \frac{1224256361964}{2542133109473} a + \frac{459441951699}{2542133109473}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 294948493.2252423 \) (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

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\zeta_{16})^+\), 4.4.346112.1, 8.8.119793516544.1, 8.8.61334280470528.1, \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed