↑

Properties

Label 16.4.27032856763...0224.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{50}\cdot 7^{4}$
Root discriminant $14.19$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $D_4^2.C_2$ (as 16T376)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 48, -64, -80, 240, -112, -208, 300, -80, -136, 144, -40, -24, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 40*x^12 + 144*x^11 - 136*x^10 - 80*x^9 + 300*x^8 - 208*x^7 - 112*x^6 + 240*x^5 - 80*x^4 - 64*x^3 + 48*x^2 - 7)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 40*x^12 + 144*x^11 - 136*x^10 - 80*x^9 + 300*x^8 - 208*x^7 - 112*x^6 + 240*x^5 - 80*x^4 - 64*x^3 + 48*x^2 - 7, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 24 x^{13} - 40 x^{12} + 144 x^{11} - 136 x^{10} - 80 x^{9} + 300 x^{8} - 208 x^{7} - 112 x^{6} + 240 x^{5} - 80 x^{4} - 64 x^{3} + 48 x^{2} - 7 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2703285676329140224=2^{50}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} - \frac{1}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{2}{11} a^{4} - \frac{1}{11} a^{3} + \frac{4}{11} a^{2} - \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{11} + \frac{4}{11} a^{10} - \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{4}{11} a^{7} - \frac{1}{11} a^{6} + \frac{3}{11} a^{5} - \frac{2}{11} a^{3} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{14} + \frac{3}{11} a^{11} - \frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{4}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{3} - \frac{2}{11} a^{2} - \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{3443} a^{15} + \frac{148}{3443} a^{14} - \frac{50}{3443} a^{13} + \frac{1}{3443} a^{12} + \frac{116}{3443} a^{11} - \frac{1479}{3443} a^{10} + \frac{134}{3443} a^{9} + \frac{72}{313} a^{8} + \frac{1156}{3443} a^{7} + \frac{153}{3443} a^{6} + \frac{54}{313} a^{5} - \frac{370}{3443} a^{4} - \frac{834}{3443} a^{3} + \frac{353}{3443} a^{2} + \frac{1593}{3443} a - \frac{1266}{3443}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 1262.34343371 \)
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

\(\Q(\sqrt{2}) \), 4.2.1024.1, 4.2.1792.1, 4.4.7168.1, 8.2.117440512.1, 8.2.117440512.2, 8.4.205520896.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$