↑

Properties

Label 16.8.80070447678...4976.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{70}\cdot 7^{14}$
Root discriminant $113.89$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:Q_8.C_2^3$ (as 16T520)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1552516, 0, -6190464, 0, -2787904, 0, -37632, 0, 69356, 0, 1344, 0, -560, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 560*x^12 + 1344*x^10 + 69356*x^8 - 37632*x^6 - 2787904*x^4 - 6190464*x^2 + 1552516)
 
gp: K = bnfinit(x^16 - 560*x^12 + 1344*x^10 + 69356*x^8 - 37632*x^6 - 2787904*x^4 - 6190464*x^2 + 1552516, 1)
 

Normalized defining polynomial

\( x^{16} - 560 x^{12} + 1344 x^{10} + 69356 x^{8} - 37632 x^{6} - 2787904 x^{4} - 6190464 x^{2} + 1552516 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(800704476782743824084831595134976=2^{70}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.89$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{168} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} + \frac{1}{12}$, $\frac{1}{168} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} + \frac{1}{12} a$, $\frac{1}{168} a^{10} - \frac{1}{6} a^{6} + \frac{1}{12} a^{2}$, $\frac{1}{168} a^{11} - \frac{1}{6} a^{7} + \frac{1}{12} a^{3}$, $\frac{1}{3864} a^{12} - \frac{1}{1288} a^{10} + \frac{1}{644} a^{8} - \frac{1}{2} a^{6} + \frac{113}{276} a^{4} + \frac{3}{92} a^{2} - \frac{31}{138}$, $\frac{1}{3864} a^{13} - \frac{1}{1288} a^{11} + \frac{1}{644} a^{9} - \frac{1}{2} a^{7} + \frac{113}{276} a^{5} + \frac{3}{92} a^{3} - \frac{31}{138} a$, $\frac{1}{320438430956112} a^{14} - \frac{1325590271}{26703202579676} a^{12} - \frac{54374685863}{40054803869514} a^{10} - \frac{230514607417}{160219215478056} a^{8} - \frac{1}{2} a^{7} - \frac{5433797917259}{22888459354008} a^{6} + \frac{1794263854147}{5722114838502} a^{4} - \frac{586437187909}{1907371612834} a^{2} - \frac{4994328248341}{11444229677004}$, $\frac{1}{28519020355093968} a^{15} + \frac{4163260921}{53406405159352} a^{13} + \frac{37578328829}{77497337921451} a^{11} + \frac{707387271182}{594146257397791} a^{9} - \frac{176017873277925}{679024294168904} a^{7} + \frac{112666842986143}{339512147084452} a^{5} - \frac{157417421011093}{509268220626678} a^{3} + \frac{16608479777191}{84878036771113} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 93406158642.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:Q_8.C_2^3$ (as 16T520):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_4:Q_8.C_2^3$
Character table for $C_4:Q_8.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.100352.1, 8.8.31581162962944.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$