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Properties

Label 16.8.94915668639...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 41^{15}$
Root discriminant $132.92$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16384, 137248, -346560, 481410, -383805, 128874, 39962, -92330, 45690, -8930, -4308, 3246, -450, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 450*x^12 + 3246*x^11 - 4308*x^10 - 8930*x^9 + 45690*x^8 - 92330*x^7 + 39962*x^6 + 128874*x^5 - 383805*x^4 + 481410*x^3 - 346560*x^2 + 137248*x - 16384)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 450*x^12 + 3246*x^11 - 4308*x^10 - 8930*x^9 + 45690*x^8 - 92330*x^7 + 39962*x^6 + 128874*x^5 - 383805*x^4 + 481410*x^3 - 346560*x^2 + 137248*x - 16384, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 450 x^{12} + 3246 x^{11} - 4308 x^{10} - 8930 x^{9} + 45690 x^{8} - 92330 x^{7} + 39962 x^{6} + 128874 x^{5} - 383805 x^{4} + 481410 x^{3} - 346560 x^{2} + 137248 x - 16384 \)

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:  \(9491566863962023381888433837890625=5^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.92$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} + \frac{1}{5} a^{5} - \frac{1}{4} a^{4} + \frac{1}{20} a^{3} + \frac{7}{40} a^{2} - \frac{3}{20} a + \frac{2}{5}$, $\frac{1}{40} a^{9} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{8} a^{3} - \frac{1}{5} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{160} a^{10} - \frac{1}{160} a^{9} - \frac{1}{80} a^{8} - \frac{9}{80} a^{7} - \frac{1}{16} a^{6} - \frac{1}{40} a^{5} + \frac{13}{160} a^{4} - \frac{9}{160} a^{3} - \frac{33}{80} a^{2} - \frac{1}{5}$, $\frac{1}{160} a^{11} + \frac{1}{160} a^{9} - \frac{1}{40} a^{7} + \frac{1}{16} a^{6} + \frac{17}{160} a^{5} - \frac{7}{40} a^{4} - \frac{11}{32} a^{3} - \frac{39}{80} a^{2} - \frac{3}{20} a + \frac{2}{5}$, $\frac{1}{640} a^{12} - \frac{1}{320} a^{11} - \frac{1}{128} a^{9} + \frac{1}{320} a^{8} - \frac{1}{160} a^{7} - \frac{49}{640} a^{6} - \frac{17}{320} a^{5} - \frac{3}{32} a^{4} - \frac{11}{640} a^{3} + \frac{29}{320} a^{2} - \frac{7}{16} a - \frac{1}{5}$, $\frac{1}{640} a^{13} - \frac{1}{640} a^{10} - \frac{1}{80} a^{9} - \frac{1}{80} a^{8} + \frac{3}{128} a^{7} + \frac{7}{160} a^{6} - \frac{19}{160} a^{5} - \frac{31}{640} a^{4} - \frac{3}{32} a^{3} - \frac{5}{32} a^{2} - \frac{9}{40} a - \frac{1}{5}$, $\frac{1}{46829209600} a^{14} - \frac{7}{46829209600} a^{13} + \frac{5874057}{23414604800} a^{12} - \frac{70488593}{46829209600} a^{11} + \frac{88014531}{46829209600} a^{10} + \frac{103036307}{23414604800} a^{9} + \frac{514212867}{46829209600} a^{8} - \frac{4068661389}{46829209600} a^{7} + \frac{528842719}{23414604800} a^{6} - \frac{10706467051}{46829209600} a^{5} + \frac{154423049}{46829209600} a^{4} + \frac{9896764981}{23414604800} a^{3} + \frac{93383}{984800} a^{2} - \frac{356894311}{1463412800} a + \frac{9058263}{22865825}$, $\frac{1}{28706305484800} a^{15} + \frac{299}{28706305484800} a^{14} - \frac{673891927}{7176576371200} a^{13} - \frac{1304827949}{28706305484800} a^{12} - \frac{26457098447}{28706305484800} a^{11} - \frac{1346729193}{1435315274240} a^{10} - \frac{238329627889}{28706305484800} a^{9} - \frac{247987313847}{28706305484800} a^{8} + \frac{498809590641}{7176576371200} a^{7} - \frac{3188269847023}{28706305484800} a^{6} + \frac{975482372003}{28706305484800} a^{5} - \frac{687816639481}{7176576371200} a^{4} - \frac{2666203970203}{7176576371200} a^{3} + \frac{259861674557}{897072046400} a^{2} - \frac{192753816047}{448536023200} a + \frac{3014206223}{14016750725}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 10137721955400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.1723025.1, 8.8.3043035529390625.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed