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Properties

Label 16.0.70426564383...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{4}\cdot 101^{12}$
Root discriminant $47.64$
Ramified primes $5, 101$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 1, 9, -89, -75, 37, -333, 2727, 333, 37, 75, -89, -9, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 9*x^13 - 89*x^12 + 75*x^11 + 37*x^10 + 333*x^9 + 2727*x^8 - 333*x^7 + 37*x^6 - 75*x^5 - 89*x^4 + 9*x^3 + x^2 + x + 1)
 
gp: K = bnfinit(x^16 - x^15 + x^14 - 9*x^13 - 89*x^12 + 75*x^11 + 37*x^10 + 333*x^9 + 2727*x^8 - 333*x^7 + 37*x^6 - 75*x^5 - 89*x^4 + 9*x^3 + x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + x^{14} - 9 x^{13} - 89 x^{12} + 75 x^{11} + 37 x^{10} + 333 x^{9} + 2727 x^{8} - 333 x^{7} + 37 x^{6} - 75 x^{5} - 89 x^{4} + 9 x^{3} + x^{2} + x + 1 \)

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:  \(704265643832481075413250625=5^{4}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.64$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{505} a^{12} - \frac{41}{505} a^{11} + \frac{31}{505} a^{10} - \frac{41}{505} a^{9} - \frac{106}{505} a^{8} + \frac{199}{505} a^{7} + \frac{72}{505} a^{6} + \frac{104}{505} a^{5} + \frac{96}{505} a^{4} - \frac{161}{505} a^{3} - \frac{171}{505} a^{2} - \frac{161}{505} a + \frac{1}{505}$, $\frac{1}{505} a^{13} - \frac{34}{505} a^{11} + \frac{18}{505} a^{10} + \frac{31}{505} a^{9} - \frac{6}{505} a^{8} - \frac{51}{505} a^{7} + \frac{127}{505} a^{6} - \frac{84}{505} a^{5} + \frac{38}{505} a^{4} + \frac{96}{505} a^{3} - \frac{1}{505} a^{2} - \frac{237}{505} a + \frac{142}{505}$, $\frac{1}{35855} a^{14} + \frac{18}{35855} a^{13} - \frac{11}{35855} a^{12} + \frac{1291}{35855} a^{11} + \frac{2987}{35855} a^{10} - \frac{361}{7171} a^{9} + \frac{6998}{35855} a^{8} - \frac{2274}{35855} a^{7} + \frac{2832}{7171} a^{6} + \frac{16371}{35855} a^{5} + \frac{17532}{35855} a^{4} + \frac{1948}{7171} a^{3} + \frac{5407}{35855} a^{2} + \frac{7828}{35855} a + \frac{15406}{35855}$, $\frac{1}{18106775} a^{15} + \frac{191}{18106775} a^{14} - \frac{11807}{18106775} a^{13} + \frac{15647}{18106775} a^{12} + \frac{205513}{3621355} a^{11} - \frac{7643}{3621355} a^{10} + \frac{667007}{18106775} a^{9} + \frac{70052}{179275} a^{8} + \frac{24236}{179275} a^{7} - \frac{4101236}{18106775} a^{6} + \frac{540478}{3621355} a^{5} - \frac{21958}{51005} a^{4} + \frac{5730256}{18106775} a^{3} + \frac{2344211}{18106775} a^{2} + \frac{1977268}{18106775} a + \frac{4078067}{18106775}$

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:  \( 1444288.72681 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.0.1030301.1, 8.0.5307600753005.1 x2, 8.0.26538003765025.2

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$