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Properties

Label 16.0.18218244060...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 941^{4}$
Root discriminant $43.78$
Ramified primes $5, 29, 941$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T392)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1049, -514, 2635, -4718, 8226, -8836, 10853, -8962, 7405, -4132, 2651, -960, 472, -104, 39, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 39*x^14 - 104*x^13 + 472*x^12 - 960*x^11 + 2651*x^10 - 4132*x^9 + 7405*x^8 - 8962*x^7 + 10853*x^6 - 8836*x^5 + 8226*x^4 - 4718*x^3 + 2635*x^2 - 514*x + 1049)
 
gp: K = bnfinit(x^16 - 4*x^15 + 39*x^14 - 104*x^13 + 472*x^12 - 960*x^11 + 2651*x^10 - 4132*x^9 + 7405*x^8 - 8962*x^7 + 10853*x^6 - 8836*x^5 + 8226*x^4 - 4718*x^3 + 2635*x^2 - 514*x + 1049, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 39 x^{14} - 104 x^{13} + 472 x^{12} - 960 x^{11} + 2651 x^{10} - 4132 x^{9} + 7405 x^{8} - 8962 x^{7} + 10853 x^{6} - 8836 x^{5} + 8226 x^{4} - 4718 x^{3} + 2635 x^{2} - 514 x + 1049 \)

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:  \(182182440601540289016015625=5^{8}\cdot 29^{6}\cdot 941^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.78$
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:  $5, 29, 941$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{116} a^{14} + \frac{1}{58} a^{13} + \frac{13}{116} a^{12} + \frac{7}{58} a^{11} - \frac{25}{116} a^{10} + \frac{11}{116} a^{9} + \frac{13}{116} a^{8} + \frac{13}{29} a^{7} - \frac{14}{29} a^{6} - \frac{11}{58} a^{5} - \frac{14}{29} a^{4} - \frac{13}{116} a^{3} + \frac{39}{116} a^{2} + \frac{41}{116} a - \frac{11}{58}$, $\frac{1}{1419855218149362082899868} a^{15} + \frac{428697048068512389835}{709927609074681041449934} a^{14} + \frac{89013025914999079003177}{1419855218149362082899868} a^{13} - \frac{172488690747500424837089}{1419855218149362082899868} a^{12} + \frac{160104688585366479307409}{709927609074681041449934} a^{11} - \frac{197419588395826875162427}{1419855218149362082899868} a^{10} + \frac{215291429103262857312689}{1419855218149362082899868} a^{9} - \frac{102418310848013708264567}{1419855218149362082899868} a^{8} - \frac{164037644847804062590187}{354963804537340520724967} a^{7} + \frac{246972974921125598097127}{1419855218149362082899868} a^{6} + \frac{428208167883022719518803}{1419855218149362082899868} a^{5} + \frac{148292796077888202839901}{709927609074681041449934} a^{4} + \frac{199986613504539867566223}{709927609074681041449934} a^{3} + \frac{30461438975861693660919}{354963804537340520724967} a^{2} + \frac{134042470825326664279821}{354963804537340520724967} a - \frac{561330696029858168579821}{1419855218149362082899868}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( 168699.738449 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.682225.1, 4.4.23525.1, 4.4.725.1, 8.0.465430950625.2, 8.0.465430950625.1, 8.8.465430950625.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$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}$
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}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
941Data not computed