↑

Properties

Label 16.0.21257640000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{14}$
Root discriminant $18.64$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2025, 0, 4050, 0, 7425, 0, 6075, 0, 2790, 0, 765, 0, 135, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 15*x^14 + 135*x^12 + 765*x^10 + 2790*x^8 + 6075*x^6 + 7425*x^4 + 4050*x^2 + 2025)
 
gp: K = bnfinit(x^16 + 15*x^14 + 135*x^12 + 765*x^10 + 2790*x^8 + 6075*x^6 + 7425*x^4 + 4050*x^2 + 2025, 1)
 

Normalized defining polynomial

\( x^{16} + 15 x^{14} + 135 x^{12} + 765 x^{10} + 2790 x^{8} + 6075 x^{6} + 7425 x^{4} + 4050 x^{2} + 2025 \)

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:  \(212576400000000000000=2^{16}\cdot 3^{12}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.64$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{45} a^{8}$, $\frac{1}{45} a^{9}$, $\frac{1}{45} a^{10}$, $\frac{1}{45} a^{11}$, $\frac{1}{540} a^{12} - \frac{1}{90} a^{8} - \frac{1}{12} a^{6} - \frac{1}{12} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{540} a^{13} - \frac{1}{90} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{5105700} a^{14} + \frac{823}{1021140} a^{12} - \frac{163}{34038} a^{10} + \frac{13}{1220} a^{8} - \frac{71}{915} a^{6} + \frac{619}{5673} a^{4} - \frac{191}{3782} a^{2} + \frac{1445}{7564}$, $\frac{1}{5105700} a^{15} + \frac{823}{1021140} a^{13} - \frac{163}{34038} a^{11} + \frac{13}{1220} a^{9} - \frac{71}{915} a^{7} + \frac{619}{5673} a^{5} - \frac{191}{3782} a^{3} + \frac{1445}{7564} a$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -\frac{197}{283650} a^{14} - \frac{901}{113460} a^{12} - \frac{5402}{85095} a^{10} - \frac{87}{305} a^{8} - \frac{2791}{3660} a^{6} - \frac{18893}{22692} a^{4} - \frac{1275}{7564} a^{2} + \frac{6313}{7564} \) (order $30$)
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:  \( 12903.7195286 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})\), 8.4.14580000000.1 x2

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

Sibling fields

Degree 8 sibling: 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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
3Data not computed
5Data not computed