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Properties

Label 16.8.19112453149...1536.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{12}\cdot 11^{8}$
Root discriminant $21.38$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (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![-11, 88, -242, 154, 494, -962, 262, 692, -461, -306, 394, -36, -110, 46, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 46*x^13 - 110*x^12 - 36*x^11 + 394*x^10 - 306*x^9 - 461*x^8 + 692*x^7 + 262*x^6 - 962*x^5 + 494*x^4 + 154*x^3 - 242*x^2 + 88*x - 11)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 + 46*x^13 - 110*x^12 - 36*x^11 + 394*x^10 - 306*x^9 - 461*x^8 + 692*x^7 + 262*x^6 - 962*x^5 + 494*x^4 + 154*x^3 - 242*x^2 + 88*x - 11, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} + 46 x^{13} - 110 x^{12} - 36 x^{11} + 394 x^{10} - 306 x^{9} - 461 x^{8} + 692 x^{7} + 262 x^{6} - 962 x^{5} + 494 x^{4} + 154 x^{3} - 242 x^{2} + 88 x - 11 \)

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:  \(1911245314971754561536=2^{24}\cdot 3^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.38$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{12} + \frac{1}{11} a^{11} - \frac{2}{11} a^{10} - \frac{1}{11} a^{9} - \frac{2}{11} a^{8} - \frac{2}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{2}$, $\frac{1}{22} a^{13} + \frac{3}{22} a^{11} - \frac{5}{22} a^{10} + \frac{2}{11} a^{8} - \frac{7}{22} a^{7} - \frac{1}{2} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{242} a^{14} - \frac{1}{121} a^{13} + \frac{3}{242} a^{12} + \frac{2}{11} a^{11} - \frac{17}{121} a^{10} + \frac{15}{242} a^{9} - \frac{13}{121} a^{8} + \frac{29}{121} a^{7} - \frac{24}{121} a^{6} + \frac{59}{242} a^{5} - \frac{40}{121} a^{4} - \frac{1}{11} a^{3} - \frac{3}{22} a^{2} - \frac{1}{22} a - \frac{3}{22}$, $\frac{1}{242} a^{15} - \frac{1}{242} a^{13} - \frac{5}{242} a^{12} - \frac{28}{121} a^{11} + \frac{23}{121} a^{10} - \frac{7}{242} a^{9} - \frac{8}{121} a^{8} + \frac{23}{121} a^{7} + \frac{31}{121} a^{6} + \frac{115}{242} a^{5} - \frac{25}{121} a^{4} - \frac{7}{22} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a + \frac{5}{22}$

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:  \( 73681.1139795 \) (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

\(\Q(\sqrt{3}) \), 4.2.6336.1, 4.4.4752.1, 4.2.1728.1, 8.4.361304064.2, 8.4.43717791744.17, 8.4.4857532416.4

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$