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Properties

Label 16.0.39542234177...1696.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 11^{8}$
Root discriminant $53.07$
Ramified primes $2, 11$
Class number $226$ (GRH)
Class group $[226]$ (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![12769, 0, 139968, 0, 244944, 0, 163296, 0, 53460, 0, 9504, 0, 936, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 48*x^14 + 936*x^12 + 9504*x^10 + 53460*x^8 + 163296*x^6 + 244944*x^4 + 139968*x^2 + 12769)
 
gp: K = bnfinit(x^16 + 48*x^14 + 936*x^12 + 9504*x^10 + 53460*x^8 + 163296*x^6 + 244944*x^4 + 139968*x^2 + 12769, 1)
 

Normalized defining polynomial

\( x^{16} + 48 x^{14} + 936 x^{12} + 9504 x^{10} + 53460 x^{8} + 163296 x^{6} + 244944 x^{4} + 139968 x^{2} + 12769 \)

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:  \(3954223417733761003417501696=2^{64}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.07$
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$$\chi_{352}(1,·)$, $\chi_{352}(67,·)$, $\chi_{352}(197,·)$, $\chi_{352}(263,·)$, $\chi_{352}(265,·)$, $\chi_{352}(331,·)$, $\chi_{352}(21,·)$, $\chi_{352}(87,·)$, $\chi_{352}(89,·)$, $\chi_{352}(155,·)$, $\chi_{352}(285,·)$, $\chi_{352}(351,·)$, $\chi_{352}(109,·)$, $\chi_{352}(175,·)$, $\chi_{352}(177,·)$, $\chi_{352}(243,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{35} a^{8} - \frac{11}{35} a^{6} + \frac{1}{7} a^{4} + \frac{12}{35} a^{2} - \frac{13}{35}$, $\frac{1}{3955} a^{9} + \frac{934}{3955} a^{7} - \frac{97}{791} a^{5} + \frac{397}{3955} a^{3} - \frac{1868}{3955} a$, $\frac{1}{3955} a^{10} + \frac{6}{791} a^{8} + \frac{1549}{3955} a^{6} - \frac{24}{565} a^{4} - \frac{851}{3955} a^{2} - \frac{1}{35}$, $\frac{1}{3955} a^{11} + \frac{1214}{3955} a^{7} - \frac{1438}{3955} a^{5} - \frac{128}{565} a^{3} + \frac{557}{3955} a$, $\frac{1}{3955} a^{12} - \frac{29}{3955} a^{8} + \frac{74}{791} a^{6} + \frac{799}{3955} a^{4} + \frac{1461}{3955} a^{2} + \frac{3}{35}$, $\frac{1}{3955} a^{13} - \frac{229}{3955} a^{7} - \frac{1401}{3955} a^{5} + \frac{1109}{3955} a^{3} + \frac{1537}{3955} a$, $\frac{1}{3955} a^{14} - \frac{3}{3955} a^{8} + \frac{68}{3955} a^{6} - \frac{1716}{3955} a^{4} + \frac{42}{565} a^{2} + \frac{9}{35}$, $\frac{1}{3955} a^{15} - \frac{31}{113} a^{7} + \frac{112}{565} a^{5} + \frac{297}{791} a^{3} - \frac{632}{3955} a$

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

Class group and class number

$C_{226}$, which has order $226$ (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:  \( 82984.7429060446 \) (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

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\zeta_{16})^+\), 4.4.247808.1, 8.8.245635219456.1, 8.0.2147483648.1, 8.0.31441308090368.8

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 ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
11Data not computed