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Properties

Label 16.8.11033416203...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{6}\cdot 29^{8}\cdot 109^{4}$
Root discriminant $31.82$
Ramified primes $5, 29, 109$
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![1, -2, -15, 10, 16, -20, -33, -124, 415, -124, -33, -20, 16, 10, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 10*x^13 + 16*x^12 - 20*x^11 - 33*x^10 - 124*x^9 + 415*x^8 - 124*x^7 - 33*x^6 - 20*x^5 + 16*x^4 + 10*x^3 - 15*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 + 10*x^13 + 16*x^12 - 20*x^11 - 33*x^10 - 124*x^9 + 415*x^8 - 124*x^7 - 33*x^6 - 20*x^5 + 16*x^4 + 10*x^3 - 15*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} + 10 x^{13} + 16 x^{12} - 20 x^{11} - 33 x^{10} - 124 x^{9} + 415 x^{8} - 124 x^{7} - 33 x^{6} - 20 x^{5} + 16 x^{4} + 10 x^{3} - 15 x^{2} - 2 x + 1 \)

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:  \(1103341620319083198765625=5^{6}\cdot 29^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.82$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{5}{12} a^{6} - \frac{5}{12} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{60} a^{10} - \frac{1}{20} a^{8} + \frac{9}{20} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{3}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{4} a - \frac{1}{15}$, $\frac{1}{60} a^{11} + \frac{1}{30} a^{9} + \frac{7}{60} a^{8} - \frac{19}{60} a^{7} + \frac{19}{60} a^{6} + \frac{11}{60} a^{5} - \frac{23}{60} a^{4} - \frac{13}{60} a^{3} + \frac{1}{6} a^{2} + \frac{4}{15} a + \frac{5}{12}$, $\frac{1}{60} a^{12} + \frac{1}{30} a^{9} + \frac{7}{60} a^{8} + \frac{1}{3} a^{7} + \frac{1}{15} a^{6} + \frac{29}{60} a^{5} + \frac{5}{12} a^{4} - \frac{19}{60} a^{3} + \frac{9}{20} a^{2} - \frac{5}{12} a - \frac{17}{60}$, $\frac{1}{300} a^{13} + \frac{1}{150} a^{11} - \frac{1}{150} a^{10} - \frac{1}{75} a^{9} - \frac{29}{300} a^{8} + \frac{19}{150} a^{7} + \frac{7}{75} a^{6} + \frac{29}{75} a^{5} + \frac{23}{150} a^{4} + \frac{133}{300} a^{3} + \frac{11}{150} a^{2} + \frac{1}{10} a - \frac{2}{25}$, $\frac{1}{4500} a^{14} + \frac{1}{750} a^{13} + \frac{8}{1125} a^{12} + \frac{7}{900} a^{11} - \frac{1}{125} a^{10} + \frac{8}{1125} a^{9} + \frac{121}{1125} a^{8} + \frac{1541}{4500} a^{7} - \frac{433}{2250} a^{6} + \frac{241}{2250} a^{5} - \frac{37}{1500} a^{4} + \frac{97}{900} a^{3} - \frac{793}{4500} a^{2} + \frac{677}{1500} a - \frac{2099}{4500}$, $\frac{1}{13500} a^{15} - \frac{1}{13500} a^{14} + \frac{1}{675} a^{13} + \frac{37}{4500} a^{12} - \frac{71}{13500} a^{11} - \frac{1}{13500} a^{10} + \frac{29}{1350} a^{9} - \frac{98}{3375} a^{8} - \frac{172}{3375} a^{7} + \frac{1021}{3375} a^{6} + \frac{1289}{2700} a^{5} - \frac{6283}{13500} a^{4} - \frac{1141}{4500} a^{3} - \frac{527}{3375} a^{2} + \frac{2359}{13500} a + \frac{5273}{13500}$

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:  \( 2241545.57795 \) (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

\(\Q(\sqrt{29}) \), 4.4.91669.1, 4.4.458345.1, 4.4.4205.1, 8.4.210080139025.1, 8.4.210080139025.2, 8.8.210080139025.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$