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Properties

Label 16.0.24759631762...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $59.51$
Ramified primes $2, 5, 7$
Class number $340$ (GRH)
Class group $[2, 170]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92236816, 0, 26353376, 0, 6588344, 0, 1613472, 0, 393764, 0, 16464, 0, 686, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 + 686*x^12 + 16464*x^10 + 393764*x^8 + 1613472*x^6 + 6588344*x^4 + 26353376*x^2 + 92236816)
 
gp: K = bnfinit(x^16 + 28*x^14 + 686*x^12 + 16464*x^10 + 393764*x^8 + 1613472*x^6 + 6588344*x^4 + 26353376*x^2 + 92236816, 1)
 

Normalized defining polynomial

\( x^{16} + 28 x^{14} + 686 x^{12} + 16464 x^{10} + 393764 x^{8} + 1613472 x^{6} + 6588344 x^{4} + 26353376 x^{2} + 92236816 \)

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:  \(24759631762948096000000000000=2^{44}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.51$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(449,·)$, $\chi_{560}(393,·)$, $\chi_{560}(139,·)$, $\chi_{560}(531,·)$, $\chi_{560}(337,·)$, $\chi_{560}(83,·)$, $\chi_{560}(281,·)$, $\chi_{560}(27,·)$, $\chi_{560}(419,·)$, $\chi_{560}(169,·)$, $\chi_{560}(363,·)$, $\chi_{560}(113,·)$, $\chi_{560}(307,·)$, $\chi_{560}(57,·)$, $\chi_{560}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{98} a^{4}$, $\frac{1}{98} a^{5}$, $\frac{1}{686} a^{6}$, $\frac{1}{686} a^{7}$, $\frac{1}{9604} a^{8}$, $\frac{1}{9604} a^{9}$, $\frac{1}{2756348} a^{10} - \frac{17}{41}$, $\frac{1}{2756348} a^{11} - \frac{17}{41} a$, $\frac{1}{38588872} a^{12} + \frac{12}{287} a^{2}$, $\frac{1}{38588872} a^{13} + \frac{12}{287} a^{3}$, $\frac{1}{270122104} a^{14} - \frac{17}{4018} a^{4}$, $\frac{1}{270122104} a^{15} - \frac{17}{4018} a^{5}$

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

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (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:  \( -\frac{1}{19294436} a^{14} - \frac{1}{787528} a^{12} - \frac{3}{98441} a^{10} - \frac{1}{1372} a^{8} - \frac{6}{343} a^{6} - \frac{1}{82} a^{4} - \frac{2}{41} a^{2} - \frac{7}{41} \) (order $10$)
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:  \( 756104.7648687996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.2508800.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.100352.1, \(\Q(\zeta_{5})\), 4.0.8000.2, 4.0.12544000.2, 4.0.12544000.1, 8.8.6294077440000.1, 8.0.64000000.2, 8.0.157351936000000.83

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$