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Properties

Label 16.8.40519959449...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 3^{4}\cdot 5^{8}\cdot 29^{8}$
Root discriminant $22.41$
Ramified primes $2, 3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -71, -32, 250, -112, 22, 33, -378, 487, -203, 12, -10, 12, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 4*x^14 + 12*x^13 - 10*x^12 + 12*x^11 - 203*x^10 + 487*x^9 - 378*x^8 + 33*x^7 + 22*x^6 - 112*x^5 + 250*x^4 - 32*x^3 - 71*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 4*x^14 + 12*x^13 - 10*x^12 + 12*x^11 - 203*x^10 + 487*x^9 - 378*x^8 + 33*x^7 + 22*x^6 - 112*x^5 + 250*x^4 - 32*x^3 - 71*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 4 x^{14} + 12 x^{13} - 10 x^{12} + 12 x^{11} - 203 x^{10} + 487 x^{9} - 378 x^{8} + 33 x^{7} + 22 x^{6} - 112 x^{5} + 250 x^{4} - 32 x^{3} - 71 x^{2} + 5 x + 1 \)

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:  \(4051995944984100000000=2^{8}\cdot 3^{4}\cdot 5^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.41$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} - \frac{3}{10} a^{9} + \frac{3}{10} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{3}{10}$, $\frac{1}{100} a^{12} + \frac{3}{100} a^{11} + \frac{1}{25} a^{10} + \frac{1}{25} a^{9} + \frac{31}{100} a^{8} + \frac{19}{100} a^{7} - \frac{31}{100} a^{6} + \frac{11}{100} a^{5} + \frac{41}{100} a^{4} - \frac{6}{25} a^{3} + \frac{19}{100} a^{2} + \frac{27}{100} a + \frac{11}{100}$, $\frac{1}{500} a^{13} + \frac{1}{500} a^{12} + \frac{2}{125} a^{11} - \frac{1}{125} a^{10} + \frac{163}{500} a^{9} + \frac{77}{500} a^{8} - \frac{219}{500} a^{7} + \frac{13}{500} a^{6} - \frac{111}{500} a^{5} - \frac{53}{250} a^{4} - \frac{223}{500} a^{3} - \frac{81}{500} a^{2} + \frac{57}{500} a - \frac{13}{125}$, $\frac{1}{1000} a^{14} - \frac{1}{1000} a^{13} + \frac{1}{1000} a^{12} + \frac{3}{200} a^{11} - \frac{49}{1000} a^{10} - \frac{469}{1000} a^{9} + \frac{59}{125} a^{8} - \frac{247}{500} a^{7} + \frac{9}{500} a^{6} - \frac{289}{1000} a^{5} - \frac{2}{125} a^{4} + \frac{87}{200} a^{3} - \frac{63}{500} a^{2} + \frac{299}{1000} a + \frac{99}{1000}$, $\frac{1}{2345000} a^{15} - \frac{573}{2345000} a^{14} - \frac{487}{2345000} a^{13} - \frac{7117}{2345000} a^{12} + \frac{58291}{2345000} a^{11} + \frac{37299}{2345000} a^{10} - \frac{56697}{117250} a^{9} + \frac{62193}{167500} a^{8} + \frac{45659}{167500} a^{7} - \frac{105853}{469000} a^{6} - \frac{54156}{293125} a^{5} + \frac{1104947}{2345000} a^{4} - \frac{62583}{1172500} a^{3} - \frac{158569}{2345000} a^{2} - \frac{745349}{2345000} a + \frac{104224}{293125}$

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:  \( 59768.0944119 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$5$5.4.2.1$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}$
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}$