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Properties

Label 16.0.68270966254...3664.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 41^{12}$
Root discriminant $411.76$
Ramified primes $2, 3, 41$
Class number $1240475776$ (GRH)
Class group $[2, 2, 2, 2, 77529736]$ (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![18984773551104, 0, 12656515700736, 0, 1884328811136, 0, 116832673728, 0, 3459851010, 0, 50107248, 0, 338004, 0, 984, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 984*x^14 + 338004*x^12 + 50107248*x^10 + 3459851010*x^8 + 116832673728*x^6 + 1884328811136*x^4 + 12656515700736*x^2 + 18984773551104)
 
gp: K = bnfinit(x^16 + 984*x^14 + 338004*x^12 + 50107248*x^10 + 3459851010*x^8 + 116832673728*x^6 + 1884328811136*x^4 + 12656515700736*x^2 + 18984773551104, 1)
 

Normalized defining polynomial

\( x^{16} + 984 x^{14} + 338004 x^{12} + 50107248 x^{10} + 3459851010 x^{8} + 116832673728 x^{6} + 1884328811136 x^{4} + 12656515700736 x^{2} + 18984773551104 \)

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:  \(682709662540737066595680412797349253873664=2^{62}\cdot 3^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $411.76$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3936=2^{5}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(3845,·)$, $\chi_{3936}(2953,·)$, $\chi_{3936}(2861,·)$, $\chi_{3936}(1877,·)$, $\chi_{3936}(409,·)$, $\chi_{3936}(985,·)$, $\chi_{3936}(2141,·)$, $\chi_{3936}(1157,·)$, $\chi_{3936}(3361,·)$, $\chi_{3936}(1969,·)$, $\chi_{3936}(173,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(3125,·)$, $\chi_{3936}(2377,·)$, $\chi_{3936}(893,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{369} a^{4}$, $\frac{1}{369} a^{5}$, $\frac{1}{1107} a^{6}$, $\frac{1}{1107} a^{7}$, $\frac{1}{1361610} a^{8} + \frac{2}{5535} a^{6} + \frac{1}{15} a^{2} + \frac{1}{5}$, $\frac{1}{2723220} a^{9} + \frac{1}{5535} a^{7} - \frac{2}{15} a^{3} + \frac{1}{10} a$, $\frac{1}{16339320} a^{10} + \frac{1}{11070} a^{6} - \frac{1}{1845} a^{4} - \frac{1}{20} a^{2} - \frac{1}{5}$, $\frac{1}{32678640} a^{11} + \frac{1}{22140} a^{7} + \frac{2}{1845} a^{5} + \frac{17}{120} a^{3} + \frac{2}{5} a$, $\frac{1}{8038945440} a^{12} + \frac{1}{5446440} a^{8} - \frac{1}{3690} a^{6} + \frac{17}{29520} a^{4} - \frac{1}{10} a^{2}$, $\frac{1}{16077890880} a^{13} + \frac{1}{10892880} a^{9} + \frac{7}{22140} a^{7} - \frac{7}{6560} a^{5} + \frac{7}{60} a^{3} - \frac{1}{2} a$, $\frac{1}{187416854877329280} a^{14} - \frac{666763}{15618071239777440} a^{12} + \frac{70627}{3627891112608} a^{10} + \frac{79543}{2645337269610} a^{8} + \frac{310819553}{688217826240} a^{6} - \frac{40691173}{57351485520} a^{4} + \frac{492253}{3760260} a^{2} + \frac{4503096}{9714005}$, $\frac{1}{374833709754658560} a^{15} - \frac{666763}{31236142479554880} a^{13} + \frac{70627}{7255782225216} a^{11} + \frac{79543}{5290674539220} a^{9} + \frac{310819553}{1376435652480} a^{7} + \frac{114732907}{114702971040} a^{5} - \frac{761167}{7520520} a^{3} + \frac{2251548}{9714005} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{77529736}$, which has order $1240475776$ (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:  \( 640752.4929130975 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{2}, \sqrt{41})\), 4.4.3442688.2, \(\Q(\zeta_{16})^+\), 8.8.11852100665344.5, 8.0.826262465891279044608.2, 8.0.826262465891279044608.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$41$41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$