↑

Properties

Label 16.0.29630251663...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![801, 1128, 2254, 2272, 2584, 1988, 1756, 984, 684, 316, 176, 80, 27, -16, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 16*x^13 + 27*x^12 + 80*x^11 + 176*x^10 + 316*x^9 + 684*x^8 + 984*x^7 + 1756*x^6 + 1988*x^5 + 2584*x^4 + 2272*x^3 + 2254*x^2 + 1128*x + 801)
 
gp: K = bnfinit(x^16 - 10*x^14 - 16*x^13 + 27*x^12 + 80*x^11 + 176*x^10 + 316*x^9 + 684*x^8 + 984*x^7 + 1756*x^6 + 1988*x^5 + 2584*x^4 + 2272*x^3 + 2254*x^2 + 1128*x + 801, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 16 x^{13} + 27 x^{12} + 80 x^{11} + 176 x^{10} + 316 x^{9} + 684 x^{8} + 984 x^{7} + 1756 x^{6} + 1988 x^{5} + 2584 x^{4} + 2272 x^{3} + 2254 x^{2} + 1128 x + 801 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{14} - \frac{1}{15} a^{13} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{4}{15} a^{9} - \frac{1}{3} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{15} a^{4} + \frac{7}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{3} a$, $\frac{1}{546902995509567652215} a^{15} - \frac{706912679770714348}{182300998503189217405} a^{14} - \frac{34207276161681801382}{546902995509567652215} a^{13} + \frac{4739638269642988279}{109380599101913530443} a^{12} + \frac{11291413139707350979}{36460199700637843481} a^{11} + \frac{19511260478284812280}{109380599101913530443} a^{10} - \frac{6079296626517956176}{42069461193043665555} a^{9} - \frac{55511120734243593857}{546902995509567652215} a^{8} - \frac{3755379232957325028}{182300998503189217405} a^{7} - \frac{49464608389897229016}{182300998503189217405} a^{6} - \frac{60406141752216513803}{546902995509567652215} a^{5} + \frac{149273145858609576716}{546902995509567652215} a^{4} - \frac{121198471744295405297}{546902995509567652215} a^{3} + \frac{267235232368081297453}{546902995509567652215} a^{2} + \frac{212927028931445591947}{546902995509567652215} a - \frac{1524115434074310538}{182300998503189217405}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 44600.4876076 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.107584000000.3, 8.4.1679360000.4, 8.4.41984000000.15

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$