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Properties

Label 16.0.83728009570...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{4}\cdot 41^{8}$
Root discriminant $27.08$
Ramified primes $2, 5, 41$
Class number $4$
Class group $[4]$
Galois group $D_4^2.C_2$ (as 16T376)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 250, 1375, 1500, 1787, -844, -1353, -1184, 132, 304, 279, 0, -21, -26, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^13 - 21*x^12 + 279*x^10 + 304*x^9 + 132*x^8 - 1184*x^7 - 1353*x^6 - 844*x^5 + 1787*x^4 + 1500*x^3 + 1375*x^2 + 250*x + 625)
 
gp: K = bnfinit(x^16 - 26*x^13 - 21*x^12 + 279*x^10 + 304*x^9 + 132*x^8 - 1184*x^7 - 1353*x^6 - 844*x^5 + 1787*x^4 + 1500*x^3 + 1375*x^2 + 250*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{13} - 21 x^{12} + 279 x^{10} + 304 x^{9} + 132 x^{8} - 1184 x^{7} - 1353 x^{6} - 844 x^{5} + 1787 x^{4} + 1500 x^{3} + 1375 x^{2} + 250 x + 625 \)

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:  \(83728009570507816960000=2^{24}\cdot 5^{4}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.08$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{2}{5} a^{9} + \frac{1}{10} a^{8} + \frac{3}{10} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{3}{10} a^{9} - \frac{1}{5} a^{8} + \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a$, $\frac{1}{50} a^{14} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{5} a^{9} + \frac{12}{25} a^{8} + \frac{2}{25} a^{7} - \frac{23}{50} a^{6} - \frac{7}{25} a^{5} + \frac{6}{25} a^{4} - \frac{7}{25} a^{3} + \frac{7}{50} a^{2} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{4466895700169751250} a^{15} + \frac{8338121592618851}{893379140033950250} a^{14} - \frac{6691831881185309}{178675828006790050} a^{13} - \frac{88328796471527401}{4466895700169751250} a^{12} - \frac{87984789220006401}{4466895700169751250} a^{11} + \frac{27788113614958499}{893379140033950250} a^{10} + \frac{2156264496911743129}{4466895700169751250} a^{9} + \frac{1482578120362910699}{4466895700169751250} a^{8} - \frac{362182777445736374}{2233447850084875625} a^{7} + \frac{705632941320741913}{2233447850084875625} a^{6} - \frac{445730449438464924}{2233447850084875625} a^{5} + \frac{846098678658933833}{2233447850084875625} a^{4} - \frac{1589072538050836883}{4466895700169751250} a^{3} + \frac{62735596732217167}{893379140033950250} a^{2} + \frac{18193680021593841}{89337914003395025} a - \frac{1445097653619638}{17867582800679005}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 30523.2610772 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 16T376):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.13448.1, 4.0.537920.1, 4.0.67240.1, 8.4.904243520.1, 8.4.57871585280.3, 8.0.289357926400.23

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{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}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$