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Properties

Label 16.0.28320098225...1441.2
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 53^{12}$
Root discriminant $51.97$
Ramified primes $7, 53$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45487, -200163, 397448, -472479, 382792, -232324, 114701, -49818, 21168, -9327, 3752, -1234, 375, -108, 22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 22*x^14 - 108*x^13 + 375*x^12 - 1234*x^11 + 3752*x^10 - 9327*x^9 + 21168*x^8 - 49818*x^7 + 114701*x^6 - 232324*x^5 + 382792*x^4 - 472479*x^3 + 397448*x^2 - 200163*x + 45487)
 
gp: K = bnfinit(x^16 - 4*x^15 + 22*x^14 - 108*x^13 + 375*x^12 - 1234*x^11 + 3752*x^10 - 9327*x^9 + 21168*x^8 - 49818*x^7 + 114701*x^6 - 232324*x^5 + 382792*x^4 - 472479*x^3 + 397448*x^2 - 200163*x + 45487, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 22 x^{14} - 108 x^{13} + 375 x^{12} - 1234 x^{11} + 3752 x^{10} - 9327 x^{9} + 21168 x^{8} - 49818 x^{7} + 114701 x^{6} - 232324 x^{5} + 382792 x^{4} - 472479 x^{3} + 397448 x^{2} - 200163 x + 45487 \)

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:  \(2832009822518079193000591441=7^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.97$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{5}{14} a^{9} - \frac{2}{7} a^{8} - \frac{1}{2} a^{6} + \frac{3}{7} a^{5} - \frac{3}{14} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{3}{14}$, $\frac{1}{14} a^{13} - \frac{3}{14} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{2} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{14} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{14} a - \frac{1}{7}$, $\frac{1}{238} a^{14} - \frac{1}{238} a^{13} + \frac{81}{238} a^{11} + \frac{13}{34} a^{10} - \frac{43}{119} a^{9} - \frac{37}{238} a^{8} - \frac{1}{238} a^{7} + \frac{20}{119} a^{6} + \frac{9}{34} a^{5} + \frac{11}{34} a^{4} + \frac{43}{119} a^{3} + \frac{39}{238} a^{2} - \frac{19}{238} a - \frac{48}{119}$, $\frac{1}{547424321626447769607586} a^{15} + \frac{241563759632345905509}{273712160813223884803793} a^{14} - \frac{7326560096139752275027}{547424321626447769607586} a^{13} + \frac{2510575694117618460143}{273712160813223884803793} a^{12} - \frac{13137184917917360920426}{39101737259031983543399} a^{11} + \frac{185442151139956295879467}{547424321626447769607586} a^{10} + \frac{119349606682719827290805}{273712160813223884803793} a^{9} + \frac{62528667482558335347987}{273712160813223884803793} a^{8} - \frac{244940925603841051835839}{547424321626447769607586} a^{7} + \frac{90407225564249052065484}{273712160813223884803793} a^{6} - \frac{97340509693909835725475}{273712160813223884803793} a^{5} + \frac{4040434821210847695081}{78203474518063967086798} a^{4} - \frac{19285445861059413494387}{273712160813223884803793} a^{3} + \frac{14564629180149368258810}{273712160813223884803793} a^{2} - \frac{92658268470305478442337}{547424321626447769607586} a + \frac{204429314162039851588791}{547424321626447769607586}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 1065003.66499 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

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

\(\Q(\sqrt{53}) \), 4.0.148877.1, 4.2.19663.1, 4.2.1042139.1, 8.4.1004087378693.1, 8.0.1004087378693.1, 8.0.1086053695321.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$