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Properties

Label 16.0.84384277134...1361.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{8}$
Root discriminant $31.29$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1472, 5600, 12032, 17400, 18208, 13910, 7837, 3130, 982, 340, 159, 56, 13, 14, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^13 + 13*x^12 + 56*x^11 + 159*x^10 + 340*x^9 + 982*x^8 + 3130*x^7 + 7837*x^6 + 13910*x^5 + 18208*x^4 + 17400*x^3 + 12032*x^2 + 5600*x + 1472)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^13 + 13*x^12 + 56*x^11 + 159*x^10 + 340*x^9 + 982*x^8 + 3130*x^7 + 7837*x^6 + 13910*x^5 + 18208*x^4 + 17400*x^3 + 12032*x^2 + 5600*x + 1472, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{13} + 13 x^{12} + 56 x^{11} + 159 x^{10} + 340 x^{9} + 982 x^{8} + 3130 x^{7} + 7837 x^{6} + 13910 x^{5} + 18208 x^{4} + 17400 x^{3} + 12032 x^{2} + 5600 x + 1472 \)

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:  \(843842771347424502531361=11^{8}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.29$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{8} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{7}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{96} a^{12} + \frac{1}{48} a^{11} + \frac{5}{96} a^{10} - \frac{1}{24} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{11}{96} a^{6} + \frac{1}{24} a^{5} + \frac{23}{96} a^{4} - \frac{19}{48} a^{3} - \frac{1}{12} a^{2} + \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} + \frac{5}{192} a^{11} - \frac{7}{192} a^{10} - \frac{1}{32} a^{8} - \frac{35}{192} a^{7} - \frac{35}{192} a^{6} + \frac{41}{192} a^{5} + \frac{13}{192} a^{4} - \frac{23}{48} a^{3} - \frac{1}{16} a^{2} - \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{21120} a^{14} - \frac{7}{10560} a^{13} - \frac{1}{192} a^{12} + \frac{7}{2640} a^{11} - \frac{183}{7040} a^{10} - \frac{227}{10560} a^{9} - \frac{2261}{21120} a^{8} - \frac{173}{1760} a^{7} - \frac{223}{1320} a^{6} - \frac{159}{880} a^{5} - \frac{2437}{21120} a^{4} + \frac{13}{55} a^{3} + \frac{511}{1056} a^{2} + \frac{217}{660} a - \frac{281}{1320}$, $\frac{1}{231286851840} a^{15} + \frac{491273}{115643425920} a^{14} - \frac{24111}{60230951} a^{13} - \frac{26517049}{38547808640} a^{12} + \frac{6201793441}{231286851840} a^{11} + \frac{44398059}{9636952160} a^{10} - \frac{8923950821}{231286851840} a^{9} + \frac{17784683}{4818476080} a^{8} - \frac{21093927709}{115643425920} a^{7} + \frac{27991128487}{115643425920} a^{6} + \frac{16676245491}{77095617280} a^{5} - \frac{2309224279}{10513038720} a^{4} + \frac{5190918305}{11564342592} a^{3} + \frac{10610795023}{28910856480} a^{2} + \frac{842449163}{4818476080} a - \frac{3550665}{20949896}$

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:  $7$
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:  \( 1861090.21257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.10769.1, 8.0.10321451129.1, 8.0.10321451129.2, 8.0.918609150481.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$